Download Trigonometric Graphs, Identities, and Equations

CHAPTER
14
Difference Formulas, p. 954
tan a 2 tan b
tan (a 2 b) 5 }}}}}}}
1 1 tan a tan b
Unit 5
Trigonometry
Trigonometric Graphs,
Identities, and Equations
Prerequisite Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906
14.1 Graph Sine, Cosine, and Tangent Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908
14.2 Translate and Reflect Trigonometric Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915
14.3 Verify Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924
Investigating Algebra: Investigating Trigonometric Identities . . . . . . . . . . . . . 923
14.4 Solve Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931
Problem Solving Workshop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 938
Mixed Review of Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 940
14.5 Write Trigonometric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 941
CBL Activity
Collect and Model Trigonometric Data . . . . . . . . . . . . . . . . . . . . 948
14.6 Apply Sum and Difference Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949
14.7 Apply Double-Angle and Half-Angle Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955
Mixed Review of Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963
ASSESSMENT
Quizzes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922, 947, 962
Chapter Summary and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964
Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969
★ Standardized Test Preparation and Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 970
"MHFCSB Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 907, 912, 917, 961
DMBTT[POFDPN
Chapter 14 Highlights
PROBLEM SOLVING
★ ASSESSMENT
• Mixed Review of Problem Solving,
940, 963
• Multiple Representations, 914, 929, 937
• Multi-Step Problems, 921, 940, 946, 954,
961, 963
• Using Alternative Methods, 938
• Real-World Problem Solving Examples,
910, 916, 927, 932, 942, 951, 957
• Standardized Test Practice Examples,
933, 956
• Multiple Choice, 913, 920, 928, 935, 936,
944, 945, 952, 959, 960
• Short Response/Extended Response,
914, 921, 922, 929, 930, 936, 940, 946,
954, 961, 962, 963
• Writing/Open-Ended, 912, 913, 919, 927,
935, 940, 944, 945, 952, 959, 963
TECHNOLOGY
At classzone.com:
• Animated Algebra, 907, 912, 917, 961
• @Home Tutor, 906, 913, 921, 923, 929,
936, 945, 953, 961
• Online Quiz, 914, 922, 930, 937, 947,
954, 962
• Electronic Function Library, 964
• State Test Practice, 940, 963, 973
Contents
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14
Trigonometric Graphs,
Identities, and Equations
14.1 Graph Sine, Cosine, and Tangent Functions
14.2 Translate and Reflect Trigonometric Graphs
14.3 Verify Trigonometric Identities
14.4 Solve Trigonometric Equations
14.5 Write Trigonometric Functions and Models
14.6 Apply Sum and Difference Formulas
14.7 Apply Double-Angle and Half-Angle Formulas
Before
In Chapter 13, you learned the following skills, which you’ll
use in Chapter 14: evaluating trigonometric functions, finding
reference angles, and evaluating inverse trigonometric functions.
Prerequisite Skills
VOCABULARY CHECK
Copy and complete the statement.
1. The sine of u is ? .
5
2. The cosine of u is ? .
3. The tangent of u is ? .
13
u
12
SKILLS CHECK
Evaluate the expression. (Review p. 866 for 14.1.)
p
5. cos }
2
4. sin 458
p
6. tan }
3
Sketch the angle. Then find its reference angle. (Review p. 866 for 14.1.)
8. 2858
7. 21658
5p
9. 2}
8
Evaluate the expression. (Review p. 875 for 14.4.)
}
Ï3
10. sin21 }
2
11. cos21 1
}
12. tan21 Ï 3
1SFSFRVJTJUFTLJMMTQSBDUJDFBUDMBTT[POFDPN
906
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Now
In Chapter 14, you will apply the big ideas listed below and reviewed in the
Chapter Summary on page 964. You will also use the key vocabulary listed below.
Big Ideas
1 Graphing trigonometric functions
2 Solving trigonometric equations
3 Applying trigonometric formulas
KEY VOCABULARY
• amplitude, p. 908
• period, p. 908
• trigonometric identity, p. 924
• periodic function, p. 908
• frequency, p. 910
• sinusoid, p. 941
• cycle, p. 908
Why?
You can use trigonometric functions to model characteristics of a projectile’s
path. For example, you can find the horizontal distance traveled by a soccer ball
using trigonometric functions.
Algebra
The animation illustrated below for Exercise 51 on page 961 helps you answer this
question: How does changing the initial speed and angle of a soccer ball kicked
from ground level affect the horizontal distance the ball travels?
5SETHEEQUATIONBELOWTODETERMINETHEMAXIMUMANGLEATWHICHTHEBALLCANBE
KICKEDATANINITIALVELOCITYOFFTSECANDSTILLTRAVELFT
X
V SINQ
s
&IRSTSUBSTITUTETHEKNOWNVALUES
XFT
VFTSEC
s SINQ
Q Ž
Ž
FT
3TART
The angle and speed at which a ball is
kicked influence the distance it travels.
#HECK!NSWER
Given a speed and distance, use the
motion equation to solve for the kick angle.
Algebra at classzone.com
Other animations for Chapter 14: pages 912, 917, and 964
907
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14.1
Graph Sine, Cosine,
and Tangent Functions
Before
You evaluated sine, cosine, and tangent functions.
Now
You will graph sine, cosine, and tangent functions.
Why?
So you can model oscillating motion, as in Ex. 31.
Key Vocabulary
• amplitude
• periodic function
• cycle
• period
• frequency
In this lesson, you will learn to graph functions of the form y 5 a sin bx and
y 5 a cos bx where a and b are positive constants and x is in radian measure.
The graphs of all sine and cosine functions are related to the graphs of the
parent functions y 5 sin x and y 5 cos x, which are shown below.
y
y 5 sin x
M51
1
amplitude: 1
range:
21 ≤ y ≤ 1
3π
22
2π
π
22
π
2
21
π
3π
2
2π x
period:
2π
m 5 21
y
M51
range:
21 ≤ y ≤ 1
y 5 cos x
amplitude: 1
22π 2 3π 2π
2
π
22
21
m 5 21
π
2
π
3π
2
2π x
period:
2π
KEY CONCEPT
For Your Notebook
Characteristics of y 5 sin x and y 5 cos x
• The domain of each function is all real numbers.
• The range of each function is 21 ≤ y ≤ 1. Therefore, the minimum
value of each function is m 5 21 and the maximum value is M 5 1.
• The amplitude of each function’s graph is half the difference of the
1 (M 2 m) 5 1 [1 2 (21)] 5 1.
maximum M and the minimum m, or }
}
2
2
• Each function is periodic, which means that its graph has a repeating
pattern. The shortest repeating portion of the graph is called a cycle.
The horizontal length of each cycle is called the period. Each graph
shown above has a period of 2π.
• The x-intercepts for y 5 sin x occur when x 5 0, 6π, 62π, 63π, . . . .
p , 6 3p , 6 5p , 6 7p , . . . .
• The x-intercepts for y 5 cos x occur when x 5 6}
}
}
}
2
2
2
2
908
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Chapter 14 Trigonometric Graphs, Identities, and Equations
10/17/05 11:11:24 AM
For Your Notebook
KEY CONCEPT
Amplitude and Period
The amplitude and period of the graphs of y 5 a sin bx and y 5 a cos bx,
where a and b are nonzero real numbers, are as follows:
2p
Period 5 }
b
Amplitude 5 a
GRAPHING KEY POINTS Each graph below shows five key x-values on the interval
2p that you can use to sketch the graphs of y 5 a sin bx and y 5 a cos bx for
0≤x≤}
b
a > 0 and b > 0. These are the x-values where the maximum and minimum values
occur and the x-intercepts.
y 1 2π
s 4 ? b , ad
y
y 5 a sin bx
s
(0, 0)
y 5 a cos bx
(0, a)
2π
,
b
d
x
s 12 ? 2πb , 0d
s 34 ? 2πb , 0d
x
s 12 ? 2πb , 2ad
s 34 ? 2πb , 2ad
EXAMPLE 1
s2πb , ad
s 14 ? 2πb , 0d
0
Graph sine and cosine functions
Graph (a) y 5 4 sin x and (b) y 5 cos 4x.
VARY CONSTANTS
Notice how changes
in a and b affect the
graphs of y 5 a sin bx
and y 5 a cos bx. When
the value of a increases,
the amplitude increases.
When the value of b
increases, the period
decreases.
Solution
2p 5 2p 5 2π.
a. The amplitude is a 5 4 and the period is }
}
1
b
y
1
4
Intercepts: (0, 0); } p 2π, 0 5 (p , 0); (2p, 0)
2
1
2
p
1 p 2π, 4 5 }
Maximum: }
1 , 42
2
14
2
3 p 2π, 24 5 3p , 24
Minimum: }
}
2
4
2 1
1
2
π
2
3π
2
π
8
3π
8
x
2p 5 2p 5 p .
b. The amplitude is a 5 1 and the period is }
}
}
2
b
4
p , 0 5 p , 0 ; 3 p p , 0 5 3p , 0
1 p}
Intercepts: }
}
2 1} 2 } }
14
2
14
8
2
2 1
8
2
2
p, 1
Maximums: (0, 1); 1 }
2
2
p , 21 5 p , 21
1 p}
Minimum: }
2 1} 2
12
✓
GUIDED PRACTICE
2
y
x
4
for Example 1
Graph the function.
1. y 5 2 cos x
2. y 5 5 sin x
3. f(x) 5 sin πx
4. g(x) 5 cos 4πx
14.1 Graph Sine, Cosine, and Tangent Functions
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EXAMPLE 2
Graph a cosine function
1 cos 2p x.
Graph y 5 }
2
SKETCH A GRAPH
Solution
After you have drawn
one complete cycle of
the graph in Example 2
on the interval 0 ≤ x ≤ 1,
you can extend the
graph by copying the
cycle as many times as
desired to the left and
right of 0 ≤ x ≤ 1.
1 and the period is 2p 5 2p 5 1.
The amplitude is a 5 }
}
}
2
2p
b
1 p 1, 0 5 1 , 0 ;
Intercepts: 1 }
2 1} 2
4
4
y
1 }34 p 1, 0 2 5 1 }34 , 0 2
1
1 ; 1, 1
Maximums: 1 0, }
2 1 }2
2
2
1
2
x
1 p 1, 2 1 5 1 , 2 1
Minimum: 1 }
}2
1} }2
2
2
2
2
MODELING WITH TRIGONOMETRIC FUNCTIONS The periodic nature of
trigonometric functions is useful for modeling oscillating motions or repeating
patterns that occur in real life. Some examples are sound waves, the motion of
a pendulum, and seasons of the year. In such applications, the reciprocal of the
period is called the frequency, which gives the number of cycles per unit of time.
EXAMPLE 3
Model with a sine function
AUDIO TEST A sound consisting of a single frequency is
called a pure tone. An audiometer produces pure tones to
test a person’s auditory functions. Suppose an audiometer
produces a pure tone with a frequency f of 2000 hertz (cycles
per second). The maximum pressure P produced from the
pure tone is 2 millipascals. Write and graph a sine model that
gives the pressure P as a function of the time t (in seconds).
Solution
STEP 1
Find the values of a and b in the model P 5 a sin bt. The maximum
pressure is 2, so a 5 2. You can use the frequency f to find b.
b
2000 5 }
2p
1
frequency 5 }
period
4000π 5 b
The pressure P as a function of time t is given by P 5 2 sin 4000πt.
15 1 .
STEP 2 Graph the model. The amplitude is a 5 2 and the period is }
}
f
2000
Intercepts: (0, 0);
P
1 ,0 5
1
1
}, 0 ; }, 0
1 }12 p }
2000 2 1 4000 2 1 2000 2
2
1 p 1 ,2 5
1
Maximum: }
}
}, 2
14
2000
2 1 8000 2
3 p 1 , 22 5
3
Minimum: }
}
}, 22
14
910
n2pe-1401.indd 910
2000
2 1 8000
1
8000
t
2
Chapter 14 Trigonometric Graphs, Identities, and Equations
10/17/05 11:11:29 AM
✓
GUIDED PRACTICE
for Examples 2 and 3
Graph the function.
1 sin πx
5. y 5 }
4
1 cos πx
6. y 5 }
3
7. f(x) 5 2 sin 3x
8. g(x) 5 3 cos 4x
9. WHAT IF? In Example 3, how would the function change if the audiometer
produced a pure tone with a frequency of 1000 hertz?
GRAPH OF Y 5 TAN X The graphs of all tangent functions are related to the graph
of the parent function y 5 tan x, which is shown below.
y
y 5 tan x
1
22π
3π
22
2π
π
π
2
22
π
x
2π
3π
2
period:
π
The function y 5 tan x has the following characteristics:
FIND ODD
MULTIPLES
p
Odd multiples of } are
2
values such as these:
p
p
61 p } 5 6}
2
2
p
3p
63 p } 5 6}
2
2
p
5p
65 p } 5 6}
2
2
p . At these
1. The domain is all real numbers except odd multiples of }
2
x-values, the graph has vertical asymptotes.
2. The range is all real numbers. So, the function y 5 tan x does not have
a maximum or minimum value, and therefore the graph of y 5 tan x
does not have an amplitude.
3. The graph has a period of π.
4. The x-intercepts of the graph occur when x 5 0, 6π, 62π, 63π, . . . .
For Your Notebook
KEY CONCEPT
Characteristics of y 5 a tan bx
The period and vertical asymptotes of the graph of y 5 a tan bx, where
a and b are nonzero real numbers, are as follows:
p .
• The period is }
b
p .
• The vertical asymptotes are at odd multiples of }
2b
GRAPHING KEY POINTS The graph at the right
shows five key x-values that can help you sketch
the graph of y 5 a tan bx for a > 0 and b > 0.
These are the x-intercept, the x-values where
the asymptotes occur, and the x-values halfway
between the x-intercept and the asymptotes. At
each halfway point, the function’s value is either
a or 2a.
y
a
π
2 2b
π
4b
π
2b
x
14.1 Graph Sine, Cosine, and Tangent Functions
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EXAMPLE 4
Graph a tangent function
Graph one period of the function y 5 2 tan 3x.
Solution
p 5 p.
The period is }
}
3
b
Intercept: (0, 0)
p 5 p , or x 5 p ;
Asymptotes: x 5 }
}
}
2p3
2b
y
6
p 5 2 p , or x 5 2 p
x 5 2}
}
}
2p3
2b
6
2
π
p ,a 5 p ,2 5 p ,2 ;
Halfway points: 1 }
2 1} 2 1} 2
4p3
4b
p
π
12
26
12
π
6
x
p
p
, 2a 2 5 1 2} , 22 2 5 1 2}, 22 2
1 2}
4p3
12
4b
"MHFCSB
✓
at classzone.com
GUIDED PRACTICE
for Example 4
Graph one period of the function.
10. y 5 3 tan x
14.1
11. y 5 tan 2x
EXERCISES
13. g(x) 5 5 tan πx
12. f(x) 5 2 tan 4x
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 5, 17, and 31
★
5 STANDARDIZED TEST PRACTICE
Exs. 2, 15, 24, 25, and 31
5 MULTIPLE REPRESENTATIONS
Ex. 32
SKILL PRACTICE
1. VOCABULARY Copy and complete: The graphs of the functions y 5 sin x
and y 5 cos x both have a(n) ? of 2π.
2. ★ WRITING Compare the domains and ranges of the functions y 5 a sin bx,
y 5 a cos bx, and y 5 a tan bx where a and b are positive constants.
EXAMPLE 1
ANALYZING FUNCTIONS Identify the amplitude and the period of the graph of
on p. 909
for Exs. 3–14
the function.
3.
4.
y
5.
y
1
y
1
π
π
8
912
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3π
8
5π
8
x
π
2π
x
1
2
3
2
5
2
x
Chapter 14 Trigonometric Graphs, Identities, and Equations
10/17/05 11:11:30 AM
GRAPHING Graph the function.
1x
6. y 5 sin }
5
2x
8. f(x) 5 cos }
5
7. y 5 4 cos x
2 sin x
10. f(x) 5 }
3
px
11. f(x) 5 sin }
2
9. y 5 sin πx
p cos x
12. y 5 }
4
13. f (x) 5 cos 24x
14. ERROR ANALYSIS Describe and correct
}3  5 1
b
Period 5 } 5 }
}
3p
2p
2p
2
the error in finding the period of the
2 x.
function y 5 sin }
3
15. ★ MULTIPLE CHOICE The graph of which function has an amplitude of 4 and
a period of 2?
A y 5 4 cos 2x
B y 5 2 sin 4x
C y 5 4 sin πx
1 πx
D y 5 2 cos }
2
GRAPHING Graph the function.
EXAMPLES
2, 3, and 4
on pp. 910–912
for Exs. 16–24
16. y 5 2 sin 8x
17. f(x) 5 4 tan x
18. y 5 3 cos πx
19. y 5 5 sin 2x
20. f(x) 5 2 tan 4x
1 πx
21. y 5 2 cos }
4
22. f(x) 5 4 tan πx
23. y 5 π cos 4πx
24. ★ MULTIPLE CHOICE Which of the following is an asymptote of the graph of
y 5 2 tan 3x?
p
A x5}
1
C x5}
B x 5 2π
6
6
p
D x 5 2}
12
25. ★ OPEN-ENDED MATH Describe a real-life situation that can be modeled by a
periodic function.
CHALLENGE Sketch the graph of the function by plotting points. Then state the
function’s domain, range, and period.
26. y 5 csc x
27. y 5 sec x
28. y 5 cot x
PROBLEM SOLVING
EXAMPLE 3
on p. 910
for Exs. 29–30
29. PENDULUMS The motion of a certain pendulum can be modeled by the
function d 5 4 cos πt where d is the pendulum’s horizontal displacement
(in inches) relative to its position at rest and t is the time (in seconds).
Graph the function. What is the greatest horizontal distance the pendulum
will travel from its position at rest?
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
30. TUNING FORKS A tuning fork produces a
sound pressure wave that can be modeled by
P 5 0.001 sin 880t
where P is the pressure (in pascals) and t is
the time (in seconds). Find the period and
frequency of this function. Then graph the
function.
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
14.1 Graph Sine, Cosine, and Tangent Functions
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31. ★ SHORT RESPONSE A buoy oscillates up and down as waves go past.
The buoy moves a total of 3.5 feet from its low point to its high point, and
then returns to its high point every 6 seconds.
a. Write an equation that gives the buoy’s vertical position y at time t if the
buoy is at its highest point when t 5 0.
b. Explain why you chose y 5 a sin bt or y 5 a cos bt for part (a).
32.
MULTIPLE REPRESENTATIONS You are standing on a bridge, 140 feet
above the ground. You look down at a car traveling away from the underpass.
u
140 ft
d
a. Writing an Equation Write an equation that gives the car’s distance d
from the base of the bridge as a function of the angle u.
b. Drawing a Graph Graph the function found in part (a). Explain how the
graph relates to the given situation.
c. Making a Table Make a table of values for the function. Use the table to
find the car’s distance from the bridge when u 5 208, 408, and 608.
33. CHALLENGE The motion of a spring can be modeled by y 5 A cos kt where
y is the spring’s vertical displacement (in feet) relative to its position at
rest, A is the initial displacement (in feet), k is a constant that measures
the elasticity of the spring, and t is the time (in seconds).
a. Suppose you have a spring whose motion can be modeled by the
function y 5 0.2 cos 6t. Find the initial displacement and the period of
the spring. Then graph the given function.
b. Graphing Calculator If a damping force is applied to the spring, the
motion of the spring can be modeled by the function y 5 0.2e24.5t cos 4t.
Graph this function. What effect does damping have on the motion?
MIXED REVIEW
PREVIEW
Graph the function. Label the vertex and the axis of symmetry. (p. 245)
Prepare for
Lesson 14.2
in Exs. 34–39.
34. y 5 2(x 2 2)2 1 1
35. y 5 5(x 2 1)2 1 7
36. f(x) 5 2(x 1 6)2 2 3
37. y 5 23(x 1 3)2 1 2
38. y 5 0.5(x 1 2)2 1 5
39. g(x) 5 22(x 1 4)2 2 3
Let f(x) 5 x 2 2 5, g(x) 5 27x, and h(x) 5 x1/3. Find the indicated value. (p. 428)
40. g(f(9))
41. h(g(3))
42. f(g(27))
43. f(h(8))
Sketch the angle. Then find its reference angle. (p. 866)
914
n2pe-1401.indd 914
44. 2508
45. 2908
46. 21458
47. 24308
13p
48. }
3
21p
49. }
4
13p
50. 2}
6
16p
51. 2}
3
PRACTICE
Lesson 14.1,
1023
Chapter 14EXTRA
Trigonometric
Graphs,for
Identities,
andp.Equations
ONLINE QUIZ at classzone.com
10/17/05 11:11:32 AM
14.2
Before
Now
Why?
Key Vocabulary
• translation, p. 123
• reflection, p. 124
• amplitude, p. 908
• period, p. 908
Translate and Reflect
Trigonometric Graphs
You graphed sine, cosine, and tangent functions.
You will translate and reflect trigonometric graphs.
So you can model predator-prey populations, as in Ex. 54.
For Your Notebook
KEY CONCEPT
Translations of Sine and Cosine Graphs
To graph y 5 a sin b(x 2 h) 1 k or y 5 a cos b(x 2 h) 1 k where a > 0 and b > 0,
follow these steps:
STEP 1
2p , the horizontal shift h, and
Identify the amplitude a, the period }
b
the vertical shift k of the graph.
STEP 2 Draw the horizontal line y 5 k, called the midline of the graph.
STEP 3 Find the five key points by translating the key points of y 5 a sin bx
or y 5 a cos bx horizontally h units and vertically k units.
STEP 4 Draw the graph through the five translated key points.
EXAMPLE 1
Graph a vertical translation
Graph y 5 2 sin 4x 1 3.
Solution
STEP 1
Identify the amplitude, period, horizontal shift, and vertical shift.
Amplitude: a 5 2
Horizontal shift: h 5 0
2p 5 2p 5 p
Period: }
}
}
4
b
Vertical shift: k 5 3
2
STEP 2 Draw the midline of the graph, y 5 3.
FIND KEY POINTS
Because the graph is
shifted up 3 units, the
y-coordinates of the
five key points will be
increased by 3.
STEP 3 Find the five key points.
On y 5 k: (0, 0 1 3) 5 (0, 3);
p
p
p
p
1 }4 , 0 1 3 2 5 1 }4 , 3 2 ; 1 }2 , 0 1 3 2 5 1 }2 , 3 2
y
p, 2 1 3 5 p, 5
Maximum: 1 }
2 1} 2
8
8
3p , 22 1 3 5 3p , 1
Minimum: }
}
1
8
2 1
8
2
STEP 4 Draw the graph through the key points.
1
π
4
14.2 Translate and Reflect Trigonometric Graphs
n2pe-1402.indd 915
π x
2
915
10/17/05 11:12:50 AM
EXAMPLE 2
Graph a horizontal translation
Graph y 5 5 cos 2(x 2 3p).
Solution
STEP 1
Identify the amplitude, period, horizontal shift, and vertical shift.
Amplitude: a 5 5
Horizontal shift: h 5 3p
2p 5 2p 5 π
Period: }
}
Vertical shift: k 5 0
2
b
FIND KEY POINTS
Because the graph is
shifted to the right 3π
units, the x-coordinates
of the five key points
will be increased by 3π.
STEP 2 Draw the midline of the graph. Because k 5 0, the midline is the x-axis.
STEP 3 Find the five key points.
y
p 1 3p, 0 5 13p , 0 ;
On y 5 k: 1 }
2 }
1
4
2
4
3p 1 3p, 0 5 15p , 0
1}
2 1}
2
4
4
1
Maximums: (0 1 3p, 5) 5 (3π, 5);
(π 1 3p, 5) 5 (4π, 5)
π
x
p 1 3p, 25 5 7p , 25
Minimum: 1 }
2 1} 2
2
2
STEP 4 Draw the graph through the key points.
EXAMPLE 3
Graph a model for circular motion
FERRIS WHEEL Suppose you are riding a Ferris wheel that turns for 180 seconds.
Your height h (in feet) above the ground at any time t (in seconds) can be
p (t 2 10) 1 90.
modeled by the equation h 5 85 sin }
20
a. Graph your height above the ground as a function of time.
b. What are your maximum and minimum heights?
Solution
2p 5 40. The wheel turns
a. The amplitude is 85 and the period is }
p
}
20
180
} 5 4.5 times in 180 seconds, so the graph below shows 4.5 cycles.
40
The five key points are (10, 90), (20, 175), (30, 90), (40, 5), and (50, 90).
h
30
10
b. Your maximum height is 90 1 85 5 175 feet and your minimum height
is 90 2 85 5 5 feet.
916
Chapter 14 Trigonometric Graphs, Identities, and Equations
180 t
✓
GUIDED PRACTICE
for Examples 1, 2, and 3
Graph the function.
p
2. y 5 3 sin 1 x 2 }
22
1. y 5 cos x 1 4
3. f(x) 5 sin (x 1 π) 2 1
REFLECTIONS You have graphed functions of the form y 5 a sin b(x 2 h) 1 k and
y 5 a cos b(x 2 h) 1 k where a > 0. To see what happens when a < 0, consider the
graphs of y 5 2sin x and y 5 2cos x.
y
y 5 2sin x
1
y
s 3π2 , 1d
sπ2 , 0d
(2π, 0)
(0, 0)
s 3π2 , 0d
x
(π, 0)
π
2
y 5 2cos x
(π, 1)
1
π
2π
(0, 21)
sπ2 , 21d
x
(2π, 21)
Notice that the graphs are reflections of the graphs of y 5 sin x and y 5 cos x
in the x-axis. In general, when a < 0 the graphs of y 5 a sin b(x 2 h) 1 k and
y 5 a cos b(x 2 h) 1 k are reflections of the graphs of y 5 a sin b(x 2 h) 1 k
and y 5 a cos b(x 2 h) 1 k, respectively, in the midline y 5 k.
EXAMPLE 4
Combine a translation and a reflection
2 x2 p .
Graph y 5 22 sin }
1 }2
2
3
Solution
STEP 1
Identify the amplitude, period, horizontal shift, and vertical shift.
p
Horizontal shift: h 5 }
Amplitude: a 5 22 5 2
2
2
p
2
p
Period: } 5 } 5 3π
Vertical shift: k 5 0
2
b
}
3
STEP 2 Draw the midline of the graph. Because k 5 0, the midline is the x-axis.
2 x2 p .
STEP 3 Find the five key points of y 5 22 sin }
1 }2
3
2
p , 0 5 p , 0 ; 3p 1 p , 0 5 (2p, 0); 3p 1 p , 0 5 7p , 0
On y 5 k: 1 0 1 }
} 2
}
1
2 1} 2 } }
2
GRAPH
REFLECTIONS
The maximum and
minimum of the
original graph become
the minimum and
maximum, respectively,
of the reflected graph.
1
2
3p 1 p , 2 5 5p , 2
Maximum: }
}
}
1
2
4
2 1
2
2
4
2
2
1
1
2
4
11p , 22 becomes 11p , 2 .
and }
}
4
4
1
2
1
2
4
2
2 1
4
2
y
the graph is reflected in the midline
5p , 2 becomes 5p , 22
y 5 0. So, }
}
4
2
9p 1 p , 22 5 11p , 22
Minimum: }
}
}
2
STEP 4 Reflect the graph. Because a < 0,
1
1
2
2
1
x
π
3π
STEP 5 Draw the graph through the key points.
"MHFCSB
at classzone.com
14.2 Translate and Reflect Trigonometric Graphs
n2pe-1402.indd 917
917
10/17/05 11:12:54 AM
TANGENT FUNCTIONS Graphing tangent functions using translations and
reflections is similar to graphing sine and cosine functions.
EXAMPLE 5
Combine a translation and a reflection
Graph y 5 23 tan x 1 5.
Solution
STEP 1
Identify the period, horizontal shift, and vertical shift.
Period: π
Horizontal shift: h 5 0
Vertical shift: k 5 5
STEP 2 Draw the midline of the graph, y 5 5.
FIND ASYMPTOTES
STEP 3 Find the asymptotes and key points of y 5 23 tan x 1 5.
Notice that the
asymptotes are not
shifted. This is because
there is no horizontal
shift.
p 5 2p; x 5 p 5 p
Asymptotes: x 5 2}
}
}
}
2p1
2p1
2
2
On y 5 k: (0, 0 1 5) 5 (0, 5)
p , 23 1 5 5 2 p , 2 ; p , 3 1 5 5 p , 8
Halfway points: 1 2}
2 1 } 2 1}
2 1} 2
4
4
4
4
STEP 4 Reflect the graph. Because a < 0,
y
the graph is reflected in the midline
p , 2 becomes 2 p , 8
y 5 5. So, 2}
}
1
2
4
1
4
2
3
p , 8 becomes p , 2 .
and 1 }
1} 2
2
4
π x
2
π
4
22
STEP 5 Draw the graph through the key
points.
EXAMPLE 6
Model with a tangent function
GLASS ELEVATOR You are standing
120 feet from the base of a 260 foot
building. You watch your friend go
down the side of the building in a
glass elevator. Write and graph a
model that gives your friend’s
distance d (in feet) from the top
of the building as a function of
the angle of elevation u.
Your friend
d
260 – d
u
You
120 ft
Solution
Use a tangent function to write an equation relating d and u.
120
120 tan u 5 260 2 d
120 tan u 2 260 5 2d
2120 tan u 1 260 5 d
Definition of tangent
Multiply each side by 120.
n2pe-1402.indd 918
180
90
Subtract 260 from each side.
Solve for d.
The graph of d 5 2120 tan u 1 260 is shown at the right.
918
Distance (ft)
opp
adj
260 2 d
tan u 5 } 5 }
d
270
0
0
20 40 60 u
Angle (degrees)
Chapter 14 Trigonometric Graphs, Identities, and Equations
10/17/05 11:12:54 AM
✓
GUIDED PRACTICE
for Examples 4, 5, and 6
Graph the function.
1x 1 2
5. y 5 23 sin }
2
p
4. y 5 2cos 1 x 1 }
22
6. f(x) 5 2tan 2x 2 1
7. WHAT IF? In Example 6, how does the model change if you are standing
150 feet from a building that is 400 feet tall?
14.2
EXERCISES
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 11, 23, and 53
★
5 STANDARDIZED TEST PRACTICE
Exs. 2, 21, 35, 48, and 54
SKILL PRACTICE
1. VOCABULARY Copy and complete: The graph of y 5 cos 2(x 2 3) is the graph
of y 5 cos 2x translated ? units to the right.
2. ★ WRITING Describe the difference between the graphs of y 5 tan x and
y 5 2tan x. How are the graphs related?
EXAMPLES
1 and 2
on pp. 915–916
for Exs. 3–21
MATCHING Match the function with its graph.
p
3. y 5 sin 2 1 x 1 }
22
4. f(x) 5 cos (x 1 π)
5. y 5 cos x 2 2
p
6. y 5 sin 1 x 1 }
42
1x 1 1
7. y 5 cos }
2
1 (x 2 π)
8. f(x) 5 sin }
2
y
A.
B.
y
C.
1
y
1
1
2π
D.
1
y
E.
π
9π x
4
π
π
4
4π x
y
π
F.
1
2π
3π x
y
1
x
x
π
4
x
3π
4
2π
GRAPHING Graph the sine or cosine function.
9. y 5 sin x 1 3
10. y 5 cos x 2 5
11. y 5 2 cos x 1 1
12. y 5 sin 3x 2 4
p
13. f(x) 5 sin 1 x 1 }
42
p
14. y 5 cos x 2 }
22
15. y 5 cos 2(x 1 π)
1 sin x 2 3p
16. f(x) 5 }
}
2
2
p 12
18. f(x) 5 cos 1 x 2 }
82
3p 2 1
19. y 5 3 cos x 1 }
4
1
1
1
2
2
1 x1 p
17. y 5 4 sin }
}2
31
2
20. y 5 sin 2(x 1 2π) 2 3
14.2 Translate and Reflect Trigonometric Graphs
n2pe-1402.indd 919
919
10/17/05 11:12:55 AM
21. ★ MULTIPLE CHOICE The graph of which function is shown?
1x 1 3
A y 5 cos }
B y 5 cos x 1 3
C y 5 cos 2x 1 3
D y 5 cos (x 1 π) 1 3
2
y
2
π
EXAMPLE 4
GRAPHING Graph the sine or cosine function.
on p. 917
for Exs. 22–33
22. f(x) 5 2sin x 1 2
1x 1 3
23. y 5 2sin }
2
24. y 5 2cos 2x 2 2
p
25. y 5 2sin 1 x 2 }
42
26. f(x) 5 2sin (x 2 π)
1x
27. y 5 22 cos }
4
28. y 5 23 cos (x 2 π) 1 4
29. y 5 2cos (x 1 π) 1 1
30. f(x) 5 1 2 3 sin (x 1 π)
3p 1 2
31. y 5 2sin x 2 }
2
p 22
32. f(x) 5 2cos 1 x 1 }
22
p
33. y 5 24 cos 2 1 x 2 }
42
1
2
x
34. ERROR ANALYSIS Describe and
1 p 2p 2 p , 2 5 p 2 p , 2 5 (0, 2)
Maximum: 1 }
2 }
1} } 2
correct the error in determining
the maximum point of the
p .
function y 5 2 sin x 2 }
1
1
4
2
2
2
2
22
35. ★ MULTIPLE CHOICE Which of the following is a maximum point of the graph
p ?
of y 5 24 cos 1 x 2 }
22
A
EXAMPLE 5
on p. 918
for Exs. 36–41
p
1 2}2 , 4 2
B (0, 4)
C
p
D (π, 4)
1 }2 , 4 2
GRAPHING Graph the tangent function.
1 tan x
36. y 5 2}
2
37. y 5 tan 2x 2 3
38. y 5 2tan 4x 1 2
p
39. y 5 2 tan 1 x 1 }
22
p
40. y 5 2tan 2 1 x 2 }
22
1 tan x 2 p
41. y 5 2}
1 }4 2
2
WRITING EQUATIONS In Exercises 42–46, write an equation of the graph
described.
42. The graph of y 5 cos 2πx translated down 4 units and left 3 units
43. The graph of y 5 3 sin x translated up 2 units and right π units
p unit and then reflected
44. The graph of y 5 5 tan x translated right }
4
in the x-axis
1 cos πx translated down 1 unit and then reflected
45. The graph of y 5 }
3
in the line y 5 21
1 sin 6x translated down 3 units and right 1 unit,
46. The graph of y 5 }
}
2
2
3
and then reflected in the line y 5 2}
2
47. REASONING Explain how you can obtain the graph of y 5 cos x by
translating the graph of y 5 sin x.
920
n2pe-1402.indd 920
5 WORKED-OUT SOLUTIONS
★
5 STANDARDIZED
TEST PRACTICE
Chapter 14 Trigonometric
Graphs, Identities, and Equations
on p. WS1
10/17/05 11:12:56 AM
48. ★ SHORT RESPONSE Explain why there is more than one tangent function
whose graph passes through the origin and has asymptotes at x 5 2π and
x 5 π.
49. CHALLENGE Find a tangent function whose graph intersects the graph of
y 5 2 1 2 sin x only at minimum points of the sine graph.
PROBLEM SOLVING
EXAMPLE 3
on p. 916
for Exs. 50–51
50. WATER WHEEL The Great Laxey wheel, located on the Isle of Man, is one of
the largest working water wheels in the world. The wheel was built in 1854
to pump water from the mines underneath it. The height h (in feet) above
the viewing platform of a bucket on the wheel can be approximated by the
function
p t 1 34.25
h 5 36.25 sin }
12
where t is time (in seconds). Graph the function. Find the diameter of the
wheel if the lowest point on the wheel is 2 feet below the viewing platform.
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
51. AUTOMOTIVE MECHANICS The pistons in an engine force the crank pins to
rotate in a circle around the center of the crankshaft. The graph shows the
height h (in inches) of a crank pin relative to the axle as a function of time t
(in seconds). Write a cosine function for the height of the crank pin.
h
3.75
t
0.01
0.02
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
52. BLOOD PRESSURE For a certain person at rest, the blood pressure P
(in millimeters of mercury) at time t (in seconds) is given by this function:
5p t
P 5 100 2 20 cos }
3
Graph the function. If one cycle is equivalent to one heartbeat, what is the
person’s pulse rate in heartbeats per minute?
EXAMPLE 6
on p. 918
for Ex. 53
53. MULTI-STEP PROBLEM You are standing 300 feet from
the base of a 200 foot cliff. Your friend is rappelling
down the cliff.
a. Write a model that gives your friend’s distance d
(in feet) from the top of the cliff as a function of
the angle of elevation u.
b. Graph the function from part (a).
c. Determine the angle of elevation if your friend
has rappelled halfway down the cliff.
14.2 Translate and Reflect Trigonometric Graphs
n2pe-1402.indd 921
921
10/17/05 11:12:57 AM
54. ★ EXTENDED RESPONSE In a particular region, the population C of coyotes
(the predator) and the population R of rabbits (the prey) can be modeled by
pt
C 5 9000 1 3000 sin }
12
pt
and R 5 20,000 1 8000 cos }
12
where t is the time in months.
a. Determine the ratio of rabbits to coyotes when t 5 0, 6, 12, and 18 months.
b. Graph both functions in the same coordinate plane.
c. Use the graphs to explain how the changes in the two populations appear
to be related.
55. CHALLENGE Suppose a Ferris wheel has a radius of 25 feet and operates at a
speed of 2 revolutions per minute. The bottom car is 5 feet above the ground.
Write a model for a person’s height h (in feet) above the ground if the value of
h is 44 feet when t 5 0.
MIXED REVIEW
PREVIEW
Simplify the rational expression, if possible. (p. 573)
Prepare for
Lesson 14.3
in Exs. 56–59.
8x 2
56. }
2
16x 2 24x 4
x2 1 x 2 6
57. }
x 2 2 7x 1 12
x 2 1 4x 2 12
58. }
x 2 2 12x 1 36
x 2 2 8x 1 15
59. }
x 2 2 6x 1 9
Find the number of combinations. (p. 690)
60.
C
10 2
61.
C
62.
8 5
C
63.
11 3
C
14 6
Find the arc length and area of a sector with the given radius r and central
angle u. (p. 859)
p
64. r 5 6 in., u 5 }
4
7p
65. r 5 3 cm, u 5 }
12
2p
66. r 5 15 m, u 5 }
3
67. r 5 10 in., u 5 1358
68. r 5 16 cm, u 5 508
69. r 5 24 m, u 5 2408
QUIZ for Lessons 14.1–14.2
Find the amplitude and the period of the graph of the function. (p. 908)
1. y 5 cos 4x
3 sin 5x
2. y 5 }
2
1 sin x
3. f(x) 5 }
4
1 cos 2πx
4. y 5 }
2
5. y 5 sin πx
px
6. g(x) 5 3 cos }
2
1 cos 3 πx (p. 908)
8. y 5 }
}
2
2
1 x (p. 908)
9. g(x) 5 2 tan }
4
Graph the function.
7. y 5 4 sin πx (p. 908)
10. f(x) 5 22 sin 3x 1 4 (p. 915)
11. y 5 cos (x 1 π) 1 2 (p. 915)
p (p. 915)
12. y 5 2tan 2 x 1 }
2
1
2
13. WINDOW WASHERS You are standing 70 feet from the base of a 250 foot
building watching a window washer lower himself to the ground. Write and
graph a model that gives the window washer’s distance d (in feet) from the
top of the building as a function of the angle of elevation u. (p. 915)
922
n2pe-1402.indd 922
PRACTICE
Lesson 14.2,
1023
Chapter 14EXTRA
Trigonometric
Graphs,for
Identities,
andp.Equations
ONLINE QUIZ at classzone.com
10/17/05 11:12:58 AM
Use before Lesson 14.3
classzone.com
Keystrokes
14.3 Investigating Trigonometric
Identities
M AT E R I A L S • graphing calculator
QUESTION
EXPLORE
How can you use a graphing calculator to verify trigonometric
identities?
Investigate a trigonometric identity
Determine whether the equation sin 2 x 1 cos2 x 5 1 is true for no x-values,
some x-values, or all x-values.
STEP 1 Enter equations
Enter the left side of the
equation as y1 and the right
side as y 2. Use the “thick”
graph style for y 2 to
distinguish the graphs.
\Y1=(sin(X))2+
(cos(X))2
Y2=1
\Y3=
\Y4=
\Y5=
\Y6=
STEP 2 Set viewing window
STEP 3 Graph equations
Set your calculator in radian
mode. Adjust the viewing
window so that the x-axis
shows 22π ≤ x ≤ 2π and the
y-axis shows 22 ≤ y ≤ 2.
Graph the equations. The
calculator first graphs
y1 5 sin 2 x 1 cos2 x and
then y 2 5 1 as a thicker line
over the graph of y1.
WINDOW
Xmin=-6.283185
Xmax=6.2831853
Xscl=1.5707963
Ymin=-2
Ymax=2
Yscl=.5
c The graphs of each side of the equation sin 2 x 1 cos2 x 5 1 are the same.
So, the equation is true for all x-values.
DR AW CONCLUSIONS
Use your observations to complete these exercises
Use a graphing calculator to determine whether the equation is true for
no x-values, some x-values, or all x-values. (Set your calculator in radian
mode and use 22p ≤ x ≤ 2p and 22 ≤ y ≤ 2 for the viewing window.)
sin x
1. tan x 5 }
cos x
2. sin x 5 2cos x
1
3. tan x 5 }
x
4. cos (23x) 5 cos 3x
5. cos x 5 1.5
6. sin (x 2 π) 5 cos x
7. sin (2x) 5 2sin x
x 5 1 cos x
8. cos }
}
2
2
p 5 sin x
9. cos 1 x 2 }
22
10. REASONING Trigonometric equations that are true for all values
of x (in their domain) are called trigonometric identities. Which
trigonometric equations in Exercises 1–9 are trigonometric identities?
14.3 Verify Trigonometric Identities
923
14.3
Before
You graphed trigonometric functions.
Now
You will verify trigonometric identities.
Why?
Key Vocabulary
• trigonometric
identity
Verify Trigonometric
Identities
So you can model the path of Halley’s comet, as in Ex. 41.
Recall from Lesson 13.3 that if an
angle u is in standard position with its
terminal side intersecting the unit circle
at (x, y), then x 5 cos u and y 5 sin u.
Because (x, y) is on a circle centered
at the origin with radius 1, it follows that:
y
(cos u, sin u ) 5 (x, y)
r51
u
x
x2 1 y 2 5 1
cos2 u 1 sin 2 u 5 1
The equation cos2 u 1 sin 2 u 5 1 is true for any value of u. A trigonometric
equation that is true for all values of u (in its domain) is called a trigonometric
identity. Several fundamental trigonometric identities are listed below, some of
which you have already learned.
For Your Notebook
KEY CONCEPT
Fundamental Trigonometric Identities
Reciprocal Identities
1
csc u 5 }
sin u
1
sec u 5 }
cos u
1
cot u 5 }
tan u
Tangent and Cotangent Identities
sin u
tan u 5 }
cos u
cos u
cot u 5 }
sin u
Pythagorean Identities
sin 2 u 1 cos2 u 5 1
1 1 tan 2 u 5 sec2 u
1 1 cot 2 u 5 csc2 u
p 2 u 5 sin u
cos 1 }
2
p 2 u 5 cot u
tan 1 }
2
cos (2u) 5 cos u
tan (2u) 5 2tan u
Cofunction Identities
p 2 u 5 cos u
sin 1 }
2
2
2
2
Negative Angle Identities
sin (2u) 5 2sin u
You can use trigonometric identities to evaluate trigonometric functions,
simplify trigonometric expressions, and verify other identities.
924
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Chapter 14 Trigonometric Graphs, Identities, and Equations
10/17/05 11:14:52 AM
EXAMPLE 1
Find trigonometric values
4 and p < u < p, find the values of the other five
Given that sin u 5 }
}
5
2
trigonometric functions of u.
Solution
STEP 1
Find cos u.
sin2 u 1 cos2 u 5 1
1 }45 2
2
Write Pythagorean identity.
4
5
1 cos2 u 5 1
Substitute } for sin u.
4
cos2 u 5 1 2 1 }
2
2
5
9
cos2 u 5 }
25
For help with finding the
sign of a trigonometric
function value, see
p. 866.
2
152
from each side.
Simplify.
3
cos u 5 6}
Take square roots of each side.
3
cos u 5 2}
Because u is in Quadrant II, cos u is negative.
5
REVIEW
TRIGONOMETRY
4
Subtract }
5
STEP 2 Find the values of the other four trigonometric functions of u using the
known values of sin u and cos u.
4
}
3
2}
5
sin u
4
5}
5 2}
tan u 5 }
3
cos u
3
2}
cos u
3
5
cot u 5 }
5 }
5 2}
4
4
sin u
}
1 5 1 55
csc u 5 }
}
}
4
4
sin u
}
1 5 1 5 25
sec u 5 }
}
}
3
cos u
3
}
5
5
EXAMPLE 2
5
2
5
Simplify a trigonometric expression
p 2 u sin u.
Simplify the expression tan 1 }
2
2
p
tan 1 } 2 u 2 sin u 5 cot u sin u
2
cos u
5 }
(sin u)
sin u
1
2
5 cos u
EXAMPLE 3
Cofunction identity
Cotangent identity
Simplify.
Simplify a trigonometric expression
1 .
Simplify the expression csc u cot 2 u 1 }
sin u
1 5 csc u cot 2 u 1 csc u
csc u cot 2 u 1 }
sin u
Reciprocal identity
5 csc u (csc2 u 2 1) 1 csc u
Pythagorean identity
3
Distributive property
3
Simplify.
5 csc u 2 csc u 1 csc u
5 csc u
14.3 Verify Trigonometric Identities
n2pe-1403.indd 925
925
10/17/05 11:14:53 AM
✓
GUIDED PRACTICE
for Examples 1, 2, and 3
Find the values of the other five trigonometric functions of u.
3 , π < u < 3p
2. sin u 5 2}
}
7
2
1, 0 < u < p
1. cos u 5 }
}
6
2
Simplify the expression.
p 2u 21
cos 1 }
2
2
5. }
1 1 sin (2u)
tan x csc x
4. }
sec x
3. sin x cot x sec x
VERIFYING IDENTITIES You can use the fundamental identities on page 924 to
verify new trigonometric identities. When verifying an identity, begin with the
expression on one side. Use algebra and trigonometric properties to manipulate
the expression until it is identical to the other side.
EXAMPLE 4
Verify a trigonometric identity
2
u 2 1 5 sin2 u.
Verify the identity sec
}
sec2 u
sec2 u 2 1
sec2 u
1
5}
2}
}
2
sec u
sec2 u
sec2 u
1
512 }
sec u
1
Write as separate fractions.
2
2
Simplify.
5 1 2 cos2 u
Reciprocal identity
5 sin 2 u
Pythagorean identity
EXAMPLE 5
Verify a trigonometric identity
cos x .
Verify the identity sec x 1 tan x 5 }
1 2 sin x
1 1 tan x
sec x 1 tan x 5 }
cos x
1 1 sin x
5}
}
Tangent identity
1 1 sin x
5}
Add fractions.
cos x
cos x
cos x
VERIFY IDENTITIES
To verify the identity,
you must introduce
1 2 sin x into the
denominator. Multiply
the numerator and
the denominator by
1 2 sin x so you get an
equivalent expression.
1 1 sin x p 1 2 sin x
5}
}
1 2 sin x
cos x
2
1 2 sin x
5}
cos x (1 2 sin x)
2
n2pe-1403.indd 926
1 2 sin x
Multiply by } .
1 2 sin x
Simplify numerator.
cos x
5}
Pythagorean identity
cos x
5}
Simplify.
cos x (1 2 sin x)
1 2 sin x
926
Reciprocal identity
Chapter 14 Trigonometric Graphs, Identities, and Equations
10/17/05 11:14:54 AM
EXAMPLE 6
Verify a real-life trigonometric identity
SHADOW LENGTH A vertical gnomon (the part of
a sundial that projects a shadow) has height h.
The length s of the shadow cast by the gnomon
when the angle of the sun above the horizon is u
can be modeled by the equation below. Show that
the equation is equivalent to s 5 h cot u.
h sin (908 2 u)
s5 }
sin u
Solution
Simplify the equation.
h sin (908 2 u)
s5 }
sin u
Write original equation.
p 2u
h sin 1 }
2
2
✓
5}
sin u
Convert 90 8 to radians.
h cos u
5}
sin u
Cofunction identity
5 h cot u
Cotangent identity
GUIDED PRACTICE
for Examples 4, 5, and 6
Verify the identity.
14.3
6. cot (2u) 5 2cot u
7. csc2 x (1 2 sin 2 x) 5 cot 2 x
8. cos x csc x tan x 5 1
9. (tan 2 x 1 1)(cos2 x 2 1) 5 2tan 2 x
EXERCISES
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 5, 11, and 41
★
5 STANDARDIZED TEST PRACTICE
Exs. 2, 9, 24, 42, 43, and 44
5 MULTIPLE REPRESENTATIONS
Ex. 41
SKILL PRACTICE
1. VOCABULARY What is a trigonometric identity?
p 2 u 5 cos u tell you
2. ★ WRITING What does the cofunction identity sin 1 }
2
2
about the graphs of y 5 sin x and y 5 cos x?
EXAMPLE 1
on p. 925
for Exs. 3–9
FINDING VALUES Find the values of the other five trigonometric functions of u.
1, 0 < u < p
3. sin u 5 }
}
3
2
3, 0 < u < p
4. tan u 5 }
}
7
2
5 , 3p < u < 2π
5. cos u 5 }
}
6 2
7 , π < u < 3p
6. sin u 5 2}
}
10
2
2, p < u < π
7. cot u 5 2}
}
5 2
9, p < u < π
8. sec u 5 2}
}
4 2
14.3 Verify Trigonometric Identities
n2pe-1403.indd 927
927
10/17/05 11:14:55 AM
3 and p < u < π, what is the value of tan u ?
9. ★ MULTIPLE CHOICE If csc u 5 }
}
2
2
}
}
}
2Ï13
B 2}
2Ï 5
A 2}
5
2Ï 13
C }
13
13
EXAMPLES
2 and 3
SIMPLIFYING EXPRESSIONS Simplify the expression.
on p. 925
for Exs. 10–24
10. sin x cot x
sin (2u)
11. }
cos (2u)
}
2Ï 5
D }
5
12. csc u sin u 1 cot 2 u
p 2x
cos 1 }
2
2
13. cos u (1 1 tan u)
p 2x
14. 1 1 tan 1 }
2
2
15. }
csc x
p 2u
cos 1 }
2
2
16. } 1 cos2 u
csc u
p 2x
sec x sin x 1 cos }
p 2 u sec u
17. sin 1 }
2
2
cos2 x
18. }
cot2 x
csc2 x 2 cot2 x
20. }
sin (2x) cot x
cos2 x tan2 (2x) 2 1
21. }
cos2 x
2
19.
12
2
2
}
1 1 sec x
ERROR ANALYSIS Describe and correct the error in simplifying the expression.
22.
23.
1 2 sin2 u 5 1 2 (1 2 cos2 u)
sin x
1
tan (2x) csc x 5 }
cos x p }
sin x
5 1 2 1 2 cos2 u
1
5}
cos x
5 2cos2 u
5 sec x
24. ★ MULTIPLE CHOICE Which of the following is the simplified form of the
expression cos u sec u ?
A tan u
EXAMPLES
4 and 5
on p. 926
for Exs. 25–34
B 1
C 2
D 1 2 sin 2 u
VERIFYING IDENTITIES Verify the identity.
25. sin x csc x 5 1
26. tan u csc u cos u 5 1
p 2u 11
cos 1 }
2
2
27. } 5 1
1 2 sin (2u)
p 2 x tan x 5 sin x
28. sin 1 }
2
2
csc2 u 2 cot2 u 5 sec2 u
29. }
1 2 sin2 u
30. 2 2 cos2 u 5 1 1 sin 2 u
31. sin x 1 cos x cot x 5 csc x
sin2 (2x)
32. }
5 cos2 x
tan2 x
1 1 cos x 1 sin x 5 2 csc x
33. }
}
1 1 cos x
sin x
sin x
34. }
5 csc x 1 cot x
1 2 cos (2x)
35. ODD AND EVEN FUNCTIONS A function f is odd if f(2x) 5 2f(x). A function
f is even if f (2x) 5 f (x). Which of the six trigonometric functions are odd?
Which are even?
VERIFYING IDENTITIES Verify the identity.
36. ln sec u 5 2ln cos u
37. ln tan u 5 ln sin u 2 ln cos u
38. CHALLENGE Use the Pythagorean identity sin 2 u 1 cos2 u 5 1 to derive the
other Pythagorean identities, 1 1 tan 2 u 5 sec2 u and 1 1 cot 2 u 5 csc2 u.
928
n2pe-1403.indd 928
5 WORKED-OUT SOLUTIONS
★
5 STANDARDIZED
Chapter 14 Trigonometric
Graphs, Identities, and Equations
on p. WS1
TEST PRACTICE
5 MULTIPLE
REPRESENTATIONS
10/17/05 11:14:56 AM
PROBLEM SOLVING
EXAMPLE 6
39. RATE OF CHANGE In calculus, it can be shown that the rate of change
of the function f(x) 5 sec x 1 cos x is given by this expression:
on p. 927
for Exs. 39–41
sec x tan x 2 sin x
Show that the expression for the rate of change can be written as sin x tan 2 x.
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
40. PHYSICAL SCIENCE Static friction is the amount of force
necessary to keep a stationary object on a flat surface
from moving. Suppose a book weighing W pounds is
lying on a ramp inclined at an angle u. The coefficient
of static friction u for the book can be found using
this equation:
uW cos u 5 W sin u
u
a. Solve the equation for u and simplify the result.
b. Use the equation from part (a) to determine what
happens to the value of u as the angle u increases
from 08 to 908.
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
41.
MULTIPLE REPRESENTATIONS The path of Halley’s comet is an ellipse with
the sun as a focus. The path can be estimated by the equation below, where r
is the comet’s distance (in astronomical units) from the sun and u is the angle
(in radians) between the horizontal major axis and the comet.
r
u
Halley’s
comet
Sun
1.069
r5}
p 2u
1 2 0.97 sin 1 }
2
2
a. Writing an Equation Simplify the equation given above.
b. Drawing a Graph Use a graphing calculator to graph the equation
from part (a).
c. Making a Table Make a table of values for the equation from part (a) in
p . Use the table to
which u starts at 0 and increases in increments of }
4
approximate the closest and farthest distance, in miles, Halley’s comet
is from the sun. (Note: 1 astronomical unit ø 93 million miles.)
42. ★ SHORT RESPONSE Use a reciprocal identity to describe what happens
to the value of sec u as the value of cos u increases. On what intervals
does this happen?
43. ★ SHORT RESPONSE Use the tangent identity to describe what happens
to the value of tan u as the value of sin u increases and the value of cos u
decreases. On what intervals does this happen?
14.3 Verify Trigonometric Identities
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929
10/17/05 11:14:58 AM
44. ★ EXTENDED RESPONSE When light traveling in a
medium (such as air) strikes the surface of a second
medium (such as water) at an angle u1, the light begins
to travel at a different angle u2. This change of direction
is defined by Snell’s law, n1 sin u1 5 n2 sin u2, where n1
and n2 are the indices of refraction for the two mediums.
Snell’s law can be derived from the equation:
n1
n2
2
2
n1
u1
u2
n2
}
} 5 }
}
Ïcot
Ïcot
u1 1 1
u2 1 1
a. Derive Simplify the equation to derive Snell’s law: n1 sin u1 5 n2 sin u2.
b. Solve If u1 5 558, u2 5 358, and n2 5 2, what is the value of n1?
c. Interpret If u1 5 u2, what must be true about the values of n1 and n2 ?
Explain when this situation would occur.
45. CHALLENGE Brewster’s angle is the angle u1, at which light reflected off
water is completely polarized, so that glare is minimized when you look at
the water with polarized sunglasses. Brewster’s angle can be found using
Snell’s law (see Exercise 44).
n1
a. Let sin 2 u2 5 }
n2 sin u1
1
2
2
1
n
2
2
2
and cos2 u2 5 }
n cos u1 .
1
Add the two equations to show that
n12
n22
n2
n1
sin 2 u1 1 }2 cos2 u1 5 1.
}
2
b. Show that the equation from part (a) can be simplified
n
2
2n
2
n
2
2n
2
2
1
2
1
to }
sin 2 u1 5 }
cos2 u1.
2
2
n1
n2
c. Solve the equation from part (b) to find Brewster’s angle:
n2
u1 5 tan21 }
n1
1 2
MIXED REVIEW
Find the product. (p. 195)
46.
F GF G
4 21
5
2
23
2
47.
F GF
22 1
7 2
1 0
1 22
5
0
9 22
G
48.
F
5 1 23
0 8
2
21 3
4
GF
1
2
3
8 27
6
0
5 26
PREVIEW
Solve the equation.
Prepare for
Lesson 14.4
in Exs. 49–54.
49. x 2 1 5x 1 4 5 0 (p. 252)
50. x 2 2 8x 1 12 5 0 (p. 252)
51. x2 1 3x 2 10 5 0 (p. 252)
52. 2x2 2 x 2 1 5 0 (p. 259)
53. 10x 2 1 17x 2 6 5 0 (p. 259)
54. 4x 2 2 12x 2 27 5 0 (p. 259)
G
Evaluate the expression without using a calculator. Give your answer in both
radians and degrees. (p. 875)
}
1
55. cos21 }
2
}
58. tan21 (2Ï 3 )
930
n2pe-1403.indd 930
}
Ï2
56. sin21 }
2
Ï3
57. sin21 2}
2
59. tan21 1
60. cos21 0
PRACTICE
Lesson 14.3,
1023
Chapter 14EXTRA
Trigonometric
Graphs,for
Identities,
andp.Equations
1
2
ONLINE QUIZ at classzone.com
10/17/05 11:14:59 AM
14.4
Before
Now
Why?
Solve Trigonometric
Equations
You verified trigonometric identities.
You will solve trigonometric equations.
So you can solve surface area problems, as in Ex. 43.
Key Vocabulary
In Lesson 14.3, you verified trigonometric identities. In this lesson, you will solve
• extraneous solution, trigonometric equations. To see the difference, consider the following:
p. 52
sin 2 x 1 cos2 x 5 1
Equation 1
sin x 5 1
Equation 2
Equation 1 is an identity because it is true for all real values of x. Equation 2,
however, is true only for some values of x. When you find these values, you are
solving the equation.
EXAMPLE 1
Solve a trigonometric equation
}
Solve 2 sin x 2 Ï 3 5 0.
Solution
First isolate sin x on one side of the equation.
}
2 sin x 2 Ï 3 5 0
Write original equation.
}
2 sin x 5 Ï3
}
Add Ï 3 to each side.
}
Ï3
sin x 5 }}
2
Divide each side by 2.
}
}
Ï3
Ï3
p.
One solution of sin x 5 }}
in the interval 0 ≤ x < 2π is x 5 sin21 }}
5}
2
2
3
p 5 2p . Moreover, because
The other solution in the interval is x 5 π 2 }
}}
3
3
y 5 sin x is periodic, there will be infinitely many solutions.
WRITE GENERAL
SOLUTION
To write the general
solution of a
trigonometric equation,
you can add multiples
of the period to all the
solutions from one
cycle.
You can use the two solutions found above to write the general solution:
p 1 2np
x5}
3
or
2p
x 5 }}
1 2np
3
(where n is any integer)
}
CHECK
Ï3
You can check the answer by graphing y 5 sin x and y 5 }}
in
2
the same coordinate plane. Then find the points where the graphs
intersect. You can see that there are infinitely many such points.
y
y5
1
3
2
x
π
2
y 5 sin x
14.4 Solve Trigonometric Equations
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10/17/05 11:15:53 AM
EXAMPLE 2
Solve a trigonometric equation in an interval
Solve 9 tan2 x 1 2 5 3 in the interval 0 ≤ x < 2p.
9 tan 2 x 1 2 5 3
Write original equation.
9 tan 2 x 5 1
Subtract 2 from each side.
1
tan 2 x 5 }
Divide each side by 9.
9
1
3
tan x 5 6}
REVIEW INVERSE
FUNCTIONS
For help with inverse
trigonometric functions,
see p. 875.
Take square roots of each side.
1 ø 0.322 and tan21 2 1 ø 20.322.
Using a calculator, you find that tan21 }
}2
1
3
3
Therefore, the general solution of the equation is:
x ø 0.322 1 nπ
or
x ø 20.322 1 nπ
(where n is any integer)
c The specific solutions in the interval 0 ≤ x < 2π are:
x ø 0.322
x ø 20.322 1 π ø 2.820
x ø 0.322 1 π ø 3.464
x ø 20.322 1 2π ø 5.961
EXAMPLE 3
Solve a real-life trigonometric equation
OCEANOGRAPHY The water depth d for the Bay of Fundy can be modeled by
p t
d 5 35 2 28 cos }}
6.2
where d is measured in feet and t is the time in hours. If t 5 0 represents
midnight, at what time(s) is the water depth 7 feet?
High tide
Low tide
ANOTHER WAY
Solution
For alternative methods
for solving the problem
in Example 3, turn
to page 938 for the
Problem Solving
Workshop.
Substitute 7 for d in the model and solve for t.
p t57
35 2 28 cos }}
Substitute 7 for d.
6.2
p t 5 228
228 cos }}
6.2
Subtract 35 from each side.
p t51
cos }}
Divide each side by 228.
p
6.2
cos u 5 1 when u 5 2np.
6.2
}}t 5 2nπ
t 5 12.4n
Solve for t.
c On the interval 0 ≤ t ≤ 24 (representing one full day), the water depth is 7 feet
when t 5 12.4(0) 5 0 (that is, at midnight) and when t 5 12.4(1) 5 12.4 (that is,
at 12:24 P.M.).
932
n2pe-1404.indd 932
Chapter 14 Trigonometric Graphs, Identities, and Equations
10/17/05 11:15:55 AM
✓
GUIDED PRACTICE
for Examples 1, 2, and 3
1. Find the general solution of the equation 2 sin x 1 4 5 5.
2. Solve the equation 3 csc2 x 5 4 in the interval 0 ≤ x < 2π.
3. OCEANOGRAPHY In Example 3, at what time(s) is the water depth 63 feet?
★
EXAMPLE 4
Standardized Test Practice
What is the general solution of sin3 x 2 4 sin x 5 0?
p 1 2nπ or x 5 3p 1 2nπ
A x5}
}}
p 1 2nπ or x 5 π 1 2nπ
B x5}
C x 5 π 1 2nπ
D x 5 2nπ or x 5 π 1 2nπ
2
2
2
Solution
sin3 x 2 4 sin x 5 0
2
sin x (sin x 2 4) 5 0
sin x (sin x 1 2)(sin x 2 2) 5 0
Write original equation.
Factor out sin x.
Factor difference of squares.
Set each factor equal to 0 and solve for x, if possible.
ELIMINATE
SOLUTIONS
Because sin x is never
less than 21 or greater
than 1, there are no
solutions of sin x 5 22
and sin x 5 2.
sin x 5 0
sin x 1 2 5 0
x 5 0 or x 5 π
sin x 2 2 5 0
sin x 5 22
sin x 5 2
The only solutions in the interval 0 ≤ x < 2π are x 5 0 and x 5 π.
The general solution is x 5 2nπ or x 5 π 1 2nπ where n is any integer.
c The correct answer is D. A B C D
EXAMPLE 5
Use the quadratic formula
Solve cos2 x 2 5 cos x 1 2 5 0 in the interval 0 ≤ x ≤ p.
Solution
Because the equation is in the form au2 1 bu 1 c 5 0 where u 5 cos x, you can
use the quadratic formula to solve for cos x.
cos2 x 2 5 cos x 1 2 5 0
Write original equation.
}}
2(25) 6 (25)2 2 4(1)(2)
2(1)
Ï
cos x 5 }}}}}}}}}}}}
Quadratic formula
}
5 6 Ï 17
5 }}}}
Simplify.
2
ø 4.56 or 0.44
21
x 5 cos
4.56
No solution
Use a calculator.
21
x 5 cos
ø 1.12
0.44
Use inverse cosine.
Use a calculator, if possible.
c In the interval 0 ≤ x ≤ π, the only solution is x ø 1.12.
14.4 Solve Trigonometric Equations
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933
10/17/05 11:15:56 AM
EXTRANEOUS SOLUTIONS When solving a trigonometric equation, it is possible
to obtain extraneous solutions. So, you should always check your solutions in the
original equation.
EXAMPLE 6
Solve an equation with an extraneous solution
Solve 1 1 cos x 5 sin x in the interval 0 ≤ x < 2p.
1 1 cos x 5 sin x
Write original equation.
(1 1 cos x)2 5 (sin x)2
REVIEW
FOIL METHOD
For help multiplying
binomials, see p. 245.
Square both sides.
1 1 2 cos x 1 cos2 x 5 sin2 x
Multiply.
1 1 2 cos x 1 cos2 x 5 1 2 cos2 x
Pythagorean identity
2 cos2 x 1 2 cos x 5 0
Quadratic form
2 cos x (cos x 1 1) 5 0
2 cos x 5 0
Factor out 2 cos x.
or cos x 1 1 5 0
Zero product property
cos x 5 0 or cos x 5 21
Solve for cos x.
p or x 5 3p .
On the interval 0 ≤ x < 2π, cos x 5 0 has two solutions: x 5 }
}}
2
2
On the interval 0 ≤ x < 2π, cos x 5 21 has one solution: x 5 π.
p , π, and 3p .
Therefore, 1 1 cos x 5 sin x has three possible solutions: x 5 }
}}
2
CHECK
2
To check the solutions, substitute them into the original equation
and simplify.
1 1 cos x 5 sin x
1 1 cos x 5 sin x
1 1 cos x 5 sin x
p 0 sin p
1 1 cos }
}
1 1 cos p 0 sin p
3p 0
3p
1 1 cos }}
sin }}
2
2
11001
2
1 1 (21) 0 0
151✓
050✓
1 Þ 21
3p is extraneous
c The apparent solution x 5 }}
y
y 5 1 1 cos x
2
because it does not check in the original
equation. The only solutions in the interval
p and x 5 π. Graphs of each
0 ≤ x < 2π are x 5 }
1
π
2
side of the original equation confirm the
solutions.
✓
GUIDED PRACTICE
2
1 1 0 0 21
x
y 5 sin x
for Examples 4, 5, and 6
Find the general solution of the equation.
4. sin3 x 2 sin x 5 0
}
5. 1 2 cos x 5 Ï 3 sin x
Solve the equation in the interval 0 ≤ x ≤ p.
6. 2 sin x 5 csc x
934
n2pe-1404.indd 934
7. tan 2 x 2 sin x tan 2 x 5 0
Chapter 14 Trigonometric Graphs, Identities, and Equations
10/17/05 11:15:57 AM
14.4
EXERCISES
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 5, 13, and 43
★
5 STANDARDIZED TEST PRACTICE
Exs. 2, 15, 36, 42, and 44
5 MULTIPLE REPRESENTATIONS
Ex. 43
SKILL PRACTICE
1. VOCABULARY What is the difference between a trigonometric equation and
a trigonometric identity?
2. ★ WRITING Describe several techniques for solving trigonometric equations.
EXAMPLE 1
on p. 931
for Exs. 3–15
CHECKING SOLUTIONS Verify that the given x-value is a solution of the equation.
3. 2 1 3 cos x 2 5 5 0, x 5 4π
4. π sec x 1 π 5 0, x 5 π
p
5. 12 sin 2 x 2 3 5 0, x 5 }
6
p
6. 5 tan3 x 2 5 5 0, x 5 }
4
p
7. 2 cos4 x 2 cos2 x 5 0, x 5 }
2
7p
8. 3 cot4 x 2 cot 2 x 2 24 5 0, x 5 }}
6
GENERAL SOLUTIONS Find the general solution of the equation.
}
9. 2 sin x 2 1 5 0
}
12. sin x 1 Ï 2 5 2sin x
}
10. Ï 3 csc x 1 2 5 0
11. 3 tan x 2 Ï 3 5 0
13. 4 cos2 x 2 3 5 0
14. 3 tan 2 x 2 9 5 0
15. ★ MULTIPLE CHOICE What is the general solution of the equation
4 sin x 5 2 sin x 1 1?
p 1 2nπ or x 5 7p 1 2nπ
A x5}
}}
p 1 nπ or x 5 5p 1 nπ
B x5}
}}
p 1 2nπ or x 5 5p 1 2nπ
C x5}
}}
p 1 nπ or x 5 7p 1 nπ
D x5}
}}
6
6
6
6
6
6
6
6
EXAMPLE 2
SOLVING EQUATIONS Solve the equation in the interval 0 ≤ x < 2p.
on p. 932
for Exs. 16–23
16. 5 1 2 sin x 2 7 5 0
17. 3 tan x 2 Ï 3 5 0
18. 3 cos x 5 cos x 2 1
19. 2 sin 2 x 2 1 5 0
20. 5 tan 2 x 2 15 5 0
21. 4 cos2 x 2 1 5 0
}
ERROR ANALYSIS Describe and correct the error in solving the equation in the
p.
interval 0 ≤ x ≤ }
2
22.
23.
1 sin x
sin2 x 5 }
22 cos x 5 21
2
1
cos x 5 2}
1
sin x 5 }
2
2
2p
x 5 }}
p
x5}
3
6
EXAMPLE 4
GENERAL SOLUTIONS Find the general solution of the equation.
on p. 933
for Exs. 24–29
24. sin x cos x 2 3 cos x 5 0
25. Ï 3 cos x tan x 2 cos x 5 0
27. 2 tan4 x 2 tan 2 x 2 15 5 0
28. Ï cos x 5 2 cos x 2 1
}
}
26. 2 sin3 x 5 sin x
}
29. 1 1 cos x 5 Ï 3 sin x
14.4 Solve Trigonometric Equations
n2pe-1404.indd 935
935
10/17/05 11:15:57 AM
EXAMPLES
5 and 6
SOLVING Solve the equation in the given interval. Check your solutions.
on pp. 933–934
for Exs. 30–35
}
30. sec x csc2 x 5 2 sec x; 0 ≤ x < 2π
31. Ï 3 cos2 x 5 cos2 x tan x; 0 ≤ x ≤ π
32. 2 sin 2 x 2 cos x 2 1 5 0; 0 ≤ x < 2π
p ≤x< p
33. sin 2 x 1 5 sin x 2 3 5 0; 2}
}
2
2
34. tan 2 x 2 3 tan x 1 2 5 0; 0 ≤ x ≤ π
35. cos x 1 sin x tan x 5 2; π ≤ x < 2π
36. ★ MULTIPLE CHOICE What are the points of intersection of the graphs of
y 5 4 sin x 1 1 and y 5 2 sin x 1 2 on the interval 0 ≤ x < 2π?
p
p
A
1 }6 , 23 2, 1 }2 , 23 2
C
, 32
1 }2 , 3 2, 1 }}
6
p
7p
p
5p
p
11p
B
, 32
1 }6 , 3 2, 1 }}
6
D
, 32
1 }6 , 3 2, 1 }}
6
INTERSECTION POINTS Find the points of intersection of the graphs of the given
functions in the interval 0 ≤ x < 2p.
37. y 5 cos2 x
}
38. y 5 9 sin 2 x
y 5 2 cos x 2 1
y 5 sin 2 x 1 8 sin x 2 2
39. y 5 Ï}
3 tan 2 x
y 5 Ï3 2 2 tan x
40. CHALLENGE A number c is a fixed point of a function f if f (c) 5 c. For example,
0 is a fixed point of f(x) 5 sin x because f(0) 5 sin 0 5 0.
a. Reasoning Use graphs to explain why the function g(x) 5 cos x has only
one fixed point.
b. Graphing Calculator Find the fixed point of g(x) 5 cos x.
PROBLEM SOLVING
EXAMPLE 3
on p. 932
for Exs. 41–42
41. WIND SPEED The average wind speed s (in miles per hour) in the Boston
p (t 1 3) 1 11.6 where t is the
Harbor can be approximated by s 5 3.38 sin }}
180
time in days, with t 5 0 representing January 1. On which days of the year is
the average wind speed 10 miles per hour?
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
42. ★ SHORT RESPONSE The number of degrees u north of due east (u > 0) or
south of due east (u < 0) that the sun rises in Cheyenne, Wyoming, can be
modeled by
2p t 2 1.4
u (t) 5 31 sin }}
1 365
2
where t is the time in days, with t 5 1 representing January 1. Use an
algebraic method to find at what day(s) the sun is 208 north of due east at
sunrise. Explain how you can use the graph of u (t) to check your answer.
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
936
n2pe-1404.indd 936
5 WORKED-OUT SOLUTIONS
★
5 STANDARDIZED
Chapter 14 Trigonometric
Graphs, Identities, and Equations
on p. WS1
TEST PRACTICE
5 MULTIPLE
REPRESENTATIONS
10/17/05 11:15:59 AM
43.
MULTIPLE REPRESENTATIONS The surface area S of a
honeycomb cell can be estimated by the equation shown
at the right. In the equation, h is the height (in inches),
s is the width of a side (in inches), and u is the angle
(in degrees) indicated in the diagram.
u
h 5 1.5
a. Using a Diagram Use the values of h and s in the diagram
to simplify the equation.
s 5 0.75
b. Making a Table Use a graphing calculator to make a table
1
}
Ï3 2 cos u
3
S 5 6hs 1 }s2 }}}}}
2
sin u
for the function from part (a). For what value(s) of u does
S 5 9 square inches?
2
c. Drawing a Graph Use a graphing calculator to graph the
function from part (a). What value of u minimizes the
surface area?
44. ★ EXTENDED RESPONSE The power P (in watts) used by a microwave oven is
the product of the voltage V (in volts) and the current I (in amperes). Suppose
the voltage and current can be modeled by
V 5 170 cos 120πt
and I 5 11.3 cos 120πt
where t is the time (in seconds).
a. Model Write the function P(t) for the power used by the microwave.
b. Solve At what times does the microwave use 375 watts of power?
c. Graphing Calculator Graph the function P(t). Describe how the graph
differs from that of a cosine function of the form y 5 a cos bt.
45. CHALLENGE Matrix multiplication can be used to rotate
a point (x, y) counterclockwise about the origin through
an angle u. The coordinates of the resulting point (x9, y9)
are determined by the matrix equation shown at the right.
a. The point (2, 3) is rotated counterclockwise about the
p . What are the
origin through an angle of }
3
coordinates of the resulting point?
b. Through what angle u must the point (6, 2) be rotated
}
}
to produce (x9, y9) 5 (3Ï3 2 1, Ï 3 1 3)?
y
u
(x 9, y 9)
F
cos u
sin u
2sin u
cos u
(x, y)
GF G F G
x
y
5
x
x9
y9
MIXED REVIEW
Perform the indicated operation(s).
x 2 1 3x 2 4 p x 2 2 3x 2 10 (p. 573)
46. }}}}}}
}}}}}}
x 2 1 5x 1 4
x 2 2 4x 2 5
2x 2 2 7x 1 3 4 3x 2 2 9x (p. 573)
47. }}}}}}
}}}}}}
x 2 2 6x 1 8
x 2 1 5x 2 36
x22
x14
48. }}}}}}
1 }}}}}}
(p. 582)
x 2 2 9x 1 20
x 2 2 3x 2 10
x 1 2 2 3 1 x 2 1 (p. 582)
49. }}}}
}}}
}}}
x14
x13
x 2 2 16
PREVIEW
Graph the function.
Prepare for
Lesson 14.5
in Exs. 50–58.
50. y 5 3 sin x (p. 908)
1 x (p. 908)
51. y 5 tan }
3
52. y 5 cos 2x (p. 908)
53. g(x) 5 3 sin 4x (p. 908)
54. y 5 cos 4x 1 3 (p. 915)
55. f(x) 5 cos x 1 9 (p. 915)
56. g(x) 5 2sin x 1 2 (p. 915)
57. y 5 tan 4(x 2 π) (p. 915)
p (p. 915)
58. y 5 2 cos 1 x 1 }
22
EXTRA PRACTICE for Lesson 14.4, p. 1023
ONLINE QUIZ at classzone.com
937
Using
ALTERNATIVE METHODS
LESSON 14.4
Another Way to Solve Example 3, page 932
MULTIPLE REPRESENTATIONS In Example 3 on page 932, you solved a
trigonometric equation algebraically. You can also solve a trigonometric
equation using a table or using a graph.
PROBLEM
OCEANOGRAPHY The water depth d for the Bay of Fundy can be modeled by
p t
d 5 35 2 28 cos }}
6.2
where d is measured in feet and t is the time in hours. If t 5 0 represents
midnight, at what time(s) is the water depth 7 feet?
METHOD 1
p t 5 7.
Using a Table The problem requires solving the equation 35 2 28 cos }}
6.2
One way to solve this equation is to make a table of values. You can use a
graphing calculator to make the table.
p x
STEP 1 Enter the function y 5 35 2 28 cos }}
6.2
into a graphing calculator. Note that
time is now represented by x and water
depth is now represented by y.
Y1=35-28cos( X/6.2)
Y2=
Y3=
Y4=
Y5=
Y6=
Y7=
STEP 2 Make a table of values for the function.
Set the table so that the x-values start
at 0 and increase in increments of 0.1.
(Be sure that the calculator is set in
radian mode.)
X
0
.1
.2
.3
.4
X=0
Y1
7
7.0359
7.1437
7.3229
7.5732
STEP 3 Scroll through the table to find all the
times x at which the water depth y is
7 feet. On the interval 0 ≤ x ≤ 24
(which represents one full day), you
can see that the function equals 7
when x is 0 and 12.4.
X
Y1
12.1 7.3229
12.2 7.1437
12.3 7.0359
12.4 7
12.5 7.0359
X=12.4
c The water depth is 7 feet when x 5 0 (that is, at midnight) and when x 5 12.4
(that is, at 12:24 P.M.).
938
n2pe-1404.indd 938
Chapter 14 Trigonometric Graphs, Identities, and Equations
10/17/05 11:16:01 AM
METHOD 2
Using a Graph Another approach is to use a graph to solve the equation
p t 5 7. You can use a graphing calculator to make the graph.
35 2 28 cos }}
6.2
p x
STEP 1 Enter the functions y 5 35 – 28 cos }}
6.2
and y 5 7 into a graphing calculator.
Again, note that time is now represented
by x and water depth is now represented
by y.
Y1=35-28cos( X/6.2)
Y2=7
Y3=
Y4=
Y5=
Y6=
Y7=
STEP 2 Graph the functions. Set your calculator
in radian mode. Adjust the viewing
window so that you can see where the
graphs intersect on the interval 0 ≤ x ≤ 24.
STEP 3 Find the intersection points of the two
graphs using the intersect feature of the
graphing calculator. On the interval
0 ≤ x ≤ 24, the graphs intersect at (0, 7)
and (12.4, 7). Because x represents the
number of hours since midnight, you
know that the water depth is 7 feet at
midnight and 12:24 P.M.
Intersection
X=12.4
Y=7
P R AC T I C E
SOLVING EQUATIONS Solve the equation
using a table and using a graph.
px 2 6 5 8
1. 20 sin }
4
px 1 6 5 2
2. 5 cos }
6
3. 210 cos 2πx 5 3
px 5 2
4. 3 1 4 sin }
2
p x 5 211
5. 215 2 10 sin }}
20
p x 2 p 1 22 5 17
6. 234 cos }
1 }} 2
5
10
7. WHAT IF? In the problem on page 938, suppose
you want to find the time(s) when the depth of
the water in the Bay of Fundy is 15 feet. Find
the time(s) using a table and using a graph.
8. WRITING Explain why the equation
2 sin x 1 3 5 0 has no solution. How does a
graph show this?
9. BUOY An ocean buoy bobs up and down as
waves travel past it. The buoy’s displacement
d (in feet) with respect to sea level can be
modeled by d 5 3 sin πt where t is the time
(in seconds). During the one second interval
0 ≤ t ≤ 1, when is the buoy 1.5 feet above sea
level? Solve the problem using a table and using
a graph.
Using Alternative Methods
n2pe-1404.indd 939
939
10/17/05 11:16:11 AM
MIXED REVIEW of Problem Solving
STATE TEST PRACTICE
classzone.com
Lessons 14.1–14.4
1. MULTI-STEP PROBLEM You put a reflector on
a spoke of your bicycle wheel. As you ride your
bicycle, the reflector’s height h (in inches)
above the ground is modeled by
h 5 13.5 1 11.5 cos 2πt
4. EXTENDED RESPONSE At an amusement
park you watch your friend go on a ride that
simulates a free-fall. You are standing 200 feet
from the base of the ride as it slowly begins to
pull your friend up to the top. The ride is 120
feet tall.
where t is the time (in seconds).
a. Graph the given function.
b. What is the frequency of the function?
d
What does the frequency represent in this
situation?
p will
c. At what time(s) on the interval 0 ≤ t ≤ }
Your friend
2
120 – d
the reflector be 8 inches above the ground?
u
2. MULTI-STEP PROBLEM You stand 80 feet from
the launch site of a hot air balloon. You look at
your friend who is traveling straight up in the
balloon to a height of 150 feet.
You
200 ft
a. Write an equation that gives the distance
d (in feet) your friend is from the top as a
function of the angle of elevation u.
b. What is the domain of the function?
c. Graph the function. Explain how the graph
relates to the given situation.
5. OPEN-ENDED Find two different cosine
functions that have (0, 6) as a maximum and
(π, 24) as a minimum.
6. GRIDDED ANSWER The number n (in millions)
a. Write and graph an equation that gives
your friend’s distance d (in feet) above
the ground as a function of the angle of
elevation u from you to your friend.
b. What is the angle of elevation when your
friend is at a height of 120 feet?
3. GRIDDED ANSWER What is the amplitude
of the graph below?
y
π
22
π
2
π
p t 2 1.09 1 113
n 5 24.5 sin 1 }}
2
210
where t is the time (in days) with t 5 1
representing January 1. What value of t
corresponds to the first day that 125 million
gallons of ice cream will be produced? Round
your answer to the nearest integer.
7. SHORT RESPONSE In calculus, it can be
(π, 1)
1
of gallons of ice cream produced in the United
States can be approximated by
2π
x
shown that the rate of change of the function
f(x) 5 2csc x 2 sin x is given by this
expression:
csc x cot x 2 cos x
a. Show that the expression for the rate of
change can be written as cos x cot 2 x.
(0, 27)
b. List the identities used to solve part (a).
940
n2pe-1404.indd 940
Chapter 14 Trigonometric Graphs, Identities, and Equations
10/17/05 11:16:14 AM
14.5
Before
Now
Why?
Key Vocabulary
• sinusoid
Write Trigonometric
Functions and Models
You graphed sine and cosine functions.
You will model data using sine and cosine functions.
So you can model the number of bicyclists, as in Ex. 26.
Graphs of sine and cosine functions are called sinusoids. One method to write a
sine or cosine function that models a sinusoid is to find the values of a, b, h, and
k for
y 5 a sin b(x 2 h) 1 k
or
y 5 a cos b(x 2 h) 1 k
2p is the period (b > 0), h is the horizontal shift, and
where a is the amplitude, }
b
k is the vertical shift.
EXAMPLE 1
Solve a multi-step problem
Write a function for the sinusoid shown below.
y
sπ8 , 5d
3
π
22
s
3π
,
8
21
d
π
2
π
x
3π
2
Solution
STEP 1
Find the maximum value M and minimum value m. From the graph,
M 5 5 and m 5 21.
STEP 2 Identify the vertical shift, k. The value of k is the mean of the maximum
5 1 (21)
2
M1m 5
4
and minimum values. The vertical shift is k 5 }
} 5 } 5 2.
2
2
So, k 5 2.
STEP 3 Decide whether the graph should be modeled by a sine or cosine
function. Because the graph crosses the midline y 5 2 on the y-axis,
the graph is a sine curve with no horizontal shift. So, h 5 0.
FIND PERIOD
Because the graph
p
repeats every } units,
2
p
the period is }.
2
p 5 2p . So, b 5 4.
STEP 4 Find the amplitude and period. The period is }
}
2
b
5 2 (21)
M
2
m
6
The amplitude is a5 } 5 } 5 } 5 3. The graph is not a
2
2
2
reflection, so a > 0. Therefore, a 5 3.
c The function is y 5 3 sin 4x 1 2.
14.5 Write Trigonometric Functions and Models
n2pe-1405.indd 941
941
10/17/05 11:17:58 AM
EXAMPLE 2
Model circular motion
JUMP ROPE At a Double Dutch competition, two people swing jump ropes as
shown in the diagram below. The highest point of the middle of each rope is
75 inches above the ground, and the lowest point is 3 inches. The rope makes
2 revolutions per second. Write a model for the height h (in feet) of a rope as a
function of the time t (in seconds) if the rope is at its lowest point when t 5 0.
75 in. above ground
3 in. above ground
Solution
STEP 1
Find the maximum and minimum values of the function. A rope’s
maximum height is 75 inches, so M 5 75. A rope’s minimum height
is 3 inches, so m 5 3.
STEP 2 Identify the vertical shift. The vertical shift for the model is:
M 1 m 5 75 1 3 5 78 5 39
k5}
}
}
2
2
2
STEP 3 Decide whether the height should be modeled by a sine or cosine
function. When t 5 0, the height is at its minimum. So, use a cosine
function whose graph is a reflection in the x-axis with no horizontal
shift (h 5 0).
STEP 4 Find the amplitude and period.
M 2 m 5 75 2 3 5 36.
The amplitude is a 5 }
}
2
2
Because the graph is a reflection, a < 0. So, a 5 236. Because a rope
is rotating at a rate of 2 revolutions per second, one revolution is
2p 5 0.5, and b 5 4π.
completed in 0.5 second. So, the period is }
b
c A model for the height of a rope is h 5 236 cos 4πt 1 39.
✓
GUIDED PRACTICE
for Examples 1 and 2
Write a function for the sinusoid.
1.
3
2.
y
y
1
(0, 2)
s 12 , 1d
2
2π
3
x
x
sπ3 , 22d
s 32 , 23d
3. WHAT IF? Describe how the model in Example 2 would change if the
lowest point of a rope is 5 inches above the ground and the highest point
is 70 inches above the ground.
942
n2pe-1405.indd 942
Chapter 14 Trigonometric Graphs, Identities, and Equations
10/17/05 11:18:00 AM
SINUSOIDAL REGRESSION Another way to model sinusoids is to use a graphing
calculator that has a sinusoidal regression feature. The advantage of this method
is that it uses all of the data points to find the model.
EXAMPLE 3
Use sinusoidal regression
ENERGY The table below shows the number of kilowatt hours K (in thousands)
used each month for a given year by a hangar at the Cape Canaveral Air Station in
Florida. The time t is measured in months, with t 5 1 representing January. Write
a trigonometric model that gives K as a function of t.
t
1
2
3
4
5
6
7
8
9
10
11
12
K
61.9
59
62
70.1
81.4
93.1
102.3
106.8
105.4
92.9
81.2
69.9
Solution
STEP 1
Enter the data in a graphing
calculator.
L1
L2
61.9
1
59
2
62
3
70.1
4
81.4
5
L1(1)=1
STEP 2 Make a scatter plot.
L3
STEP 3 Perform a sinusoidal regression,
STEP 4 Graph the model and the
because the scatter plot appears
sinusoidal.
data in the same viewing
window.
SinReg
y=a*sin(bx+c)+d
a=23.91304529
b=.5325144426
c=-2.691761344
d=82.44844029
c The model appears to be a good fit. So, a model for the data is
K 5 23.9 sin (0.533t 2 2.69) 1 82.4.
✓
GUIDED PRACTICE
for Example 3
4. METEOROLOGY Use a graphing calculator to write a sine model that
gives the average daily temperature T (in degrees Fahrenheit) for Boston,
Massachusetts, as a function of the time t (in months), where t 5 1
represents January.
t
1
2
3
4
5
6
7
8
9
10
11
12
T
29
32
39
48
59
68
74
72
65
54
45
35
14.5 Write Trigonometric Functions and Models
n2pe-1405.indd 943
943
10/17/05 11:18:01 AM
14.5
EXERCISES
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 5, 9, and 25
★
5 STANDARDIZED TEST PRACTICE
Exs. 2, 18, 19, 20, and 28
SKILL PRACTICE
1. VOCABULARY What is a sinusoid?
2. ★ WRITING Describe two methods you can use to model a sinusoid.
EXAMPLE 1
on p. 941
for Exs. 3–19
WRITING FUNCTIONS Write a function for the sinusoid.
3.
4.
sπ4 , 3d
y
y
(0, 5)
5
1
x
π
4
x
π
4
sπ4 , 25d
s 3π4 , 23d
5.
y
6.
(2, 6)
y
1
4
x
s 32 , 21d
(0, 2)
23
x
1
s 12 , 23d
ERROR ANALYSIS Describe and correct the error in finding the amplitude and
vertical shift for a sinusoid with a maximum point at (2, 10) and a minimum
point at (4, 26).
7.
8.
M2m
a 5 }
2
M1m
k5}
2
10 2 6
5}
214
5}
52
53
2
2
WRITING FUNCTIONS Write a function for the sinusoid with maximum at
point A and minimum at point B.
10. A(0, 4), B(π, 24)
π , 5 , B(0, 3)
11. A }
3
π , 8 , B(0, 26)
12. A }
6
3π , 9 , B(2π, 5)
13. A }
4
14. A(0, 5), B(6, 211)
15. A(0, 0), B(4π, 24)
π , 23 , B π , 27
16. A }
}
3
12
9. A(π, 6), B(3π, 26)
1
2
1
1
1
2
2 1
2
2
2π , 0 , B(0, 212)
17. A }
3
1
2
18. ★ MULTIPLE CHOICE During one cycle, a sinusoid has a minimum at (16, 38)
and a maximum at (24, 60). What is the amplitude of this sinusoid?
A 8
944
n2pe-1405.indd 944
B 11
C 22
D 49
Chapter 14 Trigonometric Graphs, Identities, and Equations
10/17/05 11:18:01 AM
19. ★ MULTIPLE CHOICE What is an equation of the
y
graph shown at the right?
p x 1 12
A y 5 23 cos }
p x 1 10
B y 5 25 cos }
p x 1 12
C y 5 3 sin }
6
p x 1 10
D y 5 25 sin }
6
6
(6, 15)
15
6
(0, 5)
x
3
20. ★ WRITING Any sinusoid can be modeled by both a sine function and a
cosine function. Therefore, you can choose the type of function that is more
convenient. Explain which type of function you would choose to model a
sinusoid whose y-intercept occurs at the minimum value of the function.
21. REASONING Model the sinusoid in Example 1 on page 941 with a cosine
function of the form y 5 a cos b(x 2 h) 1 k. Use identities to show that the
model you found is equivalent to the sine model in Example 1.
p, 3
22. CHALLENGE Write a sine function for the sinusoid with a minimum at 1 }
2 2
p, 8 .
and a maximum at 1 }
4 2
PROBLEM SOLVING
EXAMPLE 1
on p. 941
for Exs. 23–24
23. CIRCUITS A circuit has an alternating
V
voltage of 100 volts that peaks every
0.5 second. Use the graph shown at
the right to write a sinusoidal model
for the voltage V as a function of the
time t (in seconds).
s 18 , 100d
100
t
1
8
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
s 38 , 2100d
24. CLIMATOLOGY The graph below shows the average daily temperature of
Houston, Texas. Write a sinusoidal model for the average daily temperature T
(in degrees Fahrenheit) as a function of time t (in months).
Temperature
(8F)
Daily Temperature in Houston
T
80
(6, 84)
40 (0, 52)
0
0
2
4
6
8
10
Months since January 1
t
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
EXAMPLE 2
on p. 942
for Ex. 25
25. CIRCULAR MOTION One of the largest sewing machines in the world has a
flywheel (which turns as the machine sews) that is 5 feet in diameter. Write a
model for the height h (in feet) of the handle at the edge of the flywheel as a
function of the time t (in seconds). Assume that the wheel makes a complete
turn every 2 seconds and the handle is at its minimum height of 4 feet above
the ground when t 5 0.
14.5 Write Trigonometric Functions and Models
n2pe-1405.indd 945
945
10/17/05 11:18:03 AM
EXAMPLE 3
on p. 943
for Exs. 26–27
26. BICYCLISTS The table below shows the number of adult residents R
(in millions) in the United States who rode a bicycle during the months of
October 2001 through September 2002. The time t is measured in months,
with t 5 1 representing October 2001. Use a graphing calculator to write a
sinusoidal model that gives R as a function of t.
t
1
2
3
4
5
6
7
8
9
10
11
12
R
35
30
24
24
26
29
35
34
39
43
44
37
27. MULTI-STEP PROBLEM The table below shows the number of employees N
(in thousands) at a sporting goods company each year for eleven years. The
time t is measured in years, with t 5 1 representing the first year.
t
1
2
3
4
5
6
7
8
9
10
11
N
20.8
22.7
24.6
23.2
20
17.5
16.7
17.8
21
22
24.1
a. Model Use a graphing calculator to write a sinusoidal model that gives
N as a function of t.
b. Calculate Predict the number of employees in the twelfth year.
28. ★ EXTENDED RESPONSE The low tide at Eastport, Maine, is 3.5 feet
and occurs at midnight. After 6 hours, Eastport is at high tide, which is
16.5 feet.
a. Model Write a sinusoidal model that gives the tide depth d (in feet) as a
function of the time t (in hours). Let t 5 0 represent midnight.
b. Calculate Find all the times when low and high tides occur in a 24 hour
period.
c. Reasoning Explain how the graph of the function you wrote in part (a)
is related to a graph that shows the tide depth d at Eastport t hours after
3:00 A.M.
29. CHALLENGE The table below shows the average monthly sea temperatures T
(in degrees Celsius) for Santa Barbara, California. The time t is measured in
months, with t 5 1 representing January.
t
1
2
3
4
5
6
7
8
9
10
11
12
T
14
13.6
13.4
12.5
13.9
15.6
16.8
17.2
17.7
17.1
15.5
14.1
a. Use a graphing calculator to write a sine model that gives T as a
function of t.
b. Find a cosine model for the data.
946
n2pe-1405.indd 946
5 WORKED-OUT SOLUTIONS
★
5 STANDARDIZED
Chapter 14 Trigonometric
Graphs, Identities, and Equations
on p. WS1
TEST PRACTICE
10/17/05 11:18:03 AM
MIXED REVIEW
Write the standard form of the equation of the parabola with the given focus
and vertex at (0, 0). (p. 620)
30. (6, 0)
31. (24, 0)
32. (0, 23)
9
33. 0, }
4
5, 0
34. }
8
7
35. 0, 2}
12
1
PREVIEW
1
2
2
1
2
Evaluate the function without using a calculator. (p. 866)
Prepare for
Lesson 14.6
in Exs. 36–41.
36. cos (22108)
37. tan 3158
3p
39. sin }
2
11p
40. tan 2}
6
1
38. sin (21208)
4p
41. cos }
3
2
Find the area of n ABC with the given side lengths and included angle. (p. 882)
42. A 5 368, b 5 8, c 5 5
43. A 5 798, b 5 14, c 5 8
44. C 5 1458, a 5 4, b 5 14
45. B 5 508, a 5 7, c 5 9
46. A 5 1238, b 5 26, c 5 31
47. C 5 1058, a 5 22, b 5 19
QUIZ for Lessons 14.3–14.5
Simplify the expression. (p. 924)
2
p 2 u cot u 2 csc2 u
3. tan 1 }
2
2
2. sin u (1 1 cot 2 u)
1. sin x sec x
2
2
4. cos u 1 sin u 1 tan u
p 2 x sec x
tan 1 }
2
cos (2x)
sin (2x)
6. }
sec x
csc x 1 }
2
5. }
1 2 csc2 x
Find the general solution of the equation. (p. 931)
}
8. Ï 2 cos x sin x 2 cos x 5 0
7. cos x 1 cos (2x) 5 1
9. 2 sin 2 x 2 sin x 5 1
Write a function for the sinusoid. (p. 941)
10.
y
1
11.
(0.5, 1)
y
(3, 14)
x
1
3
(0, 6)
x
1
(1.5, 25)
12. DAILY TEMPERATURES The table below shows the average daily
temperature D (in degrees Fahrenheit) in Detroit, Michigan. The time t
is measured in months, with t 5 1 representing January. Use a graphing
calculator to write a sinusoidal model that gives D as a function of t. (p. 941)
t
1
2
3
4
5
6
7
8
9
10
11
12
D
24.5
27.2
36.9
48.1
59.8
69
73.5
71.8
63.9
51.9
40.7
29.6
EXTRA PRACTICE for Lesson 14.5, p. 1023
n2pe-1405.indd 947
ONLINE
at classzone.com
14.5 QUIZ
Write Trigonometric
Models
947
10/17/05 11:18:04 AM
Use after Lesson 14.5
classzone.com
Keystrokes
14.5 Collect and Model
Trigonometric Data
M AT E R I A L S • musical instrument • Calculator Based Laboratory (CBL)
• CBL microphone
• graphing calculator
QUESTION
How is music related to trigonometry?
Sound is a variation in pressure transmitted through air, water, or other matter.
Sound travels as a wave. The sound of a pure note can be represented using a sine
(or cosine) wave. More complicated sounds can be modeled by the sum of several
sine waves.
EXPLORE
Analyze the sound of a musical instrument
Play a note on a musical instrument. Write a sine function to describe the note.
STEP 1 Play note
Play a pure note on a musical instrument. Use the
CBL and the CBL microphone to collect the sound
data and store it in a graphing calculator.
STEP 2 Graph function
Use the graphing calculator to graph the pressure
of the sound as a function of time.
STEP 3 Find characteristics of graph
Use the graph of the sound data to calculate the
note’s amplitude and frequency (the number of
cycles in one second).
STEP 4 Write function
Write a sine function for the note.
DR AW CONCLUSIONS
Use your observations to complete these exercises
1. Choose a note to play and have a classmate also choose a note. Find two
sine functions y 5 f (x) and y 5 g(x) that model the two notes. Then play
the notes simultaneously and use the CBL and a graphing calculator to
graph the resulting sound wave. Compare this graph with the graph of
y 5 f(x) 1 g(x). What do you notice?
2. The pitch of a sound wave is determined by the wave’s frequency.
The greater the frequency, the higher the pitch. Which of the notes in
Exercise 1 had a higher pitch?
3. When you change the volume of a note, what happens to the graph of the
sound wave?
4. Compare the sine waves for different instruments playing the same note.
948
Chapter 14 Trigonometric Graphs, Identities, and Equations
14.6
Before
Now
Why?
Key Vocabulary
• trigonometric
identity, p. 924
Apply Sum and
Difference Formulas
You found trigonometric functions of a given angle.
You will use trigonometric sum and difference formulas.
So you can simplify a ratio used for aerial photography, as in Ex. 43.
In this lesson, you will study formulas that allow you to evaluate trigonometric
functions of the sum or difference of two angles.
For Your Notebook
KEY CONCEPT
Sum and Difference Formulas
Sum Formulas
Difference Formulas
sin (a 1 b) 5 sin a cos b 1 cos a sin b
sin (a 2 b) 5 sin a cos b 2 cos a sin b
cos (a 1 b) 5 cos a cos b 2 sin a sin b
cos (a 2 b) 5 cos a cos b 1 sin a sin b
tan a 1 tan b
tan (a 1 b) 5 }
tan a 2 tan b
tan (a 2 b) 5 }
1 2 tan a tan b
1 1 tan a tan b
In general, sin (a 1 b) Þ sin a 1 sin b. Similar statements can be made for the
other trigonometric functions of sums and differences.
EXAMPLE 1
Evaluate a trigonometric expression
7p
Find the exact value of (a) sin 158 and (b) tan }
.
12
a. sin 158 5 sin (608 2 458)
Substitute 60 8 2 45 8 for 15 8.
5 sin 608 cos 458 2 cos 608 sin 458
}
}
Ï3 Ï2 2 1 Ï2
5}
}
} }
2
122
}
Difference formula for sine
}
2
122
Evaluate.
}
Ï6 2 Ï2
5}
Simplify.
4
7p
p
p
b. tan } 5 tan 1 } 1 } 2
12
3
4
p
p
3
4
7p
12
Substitute } 1 } for }.
p 1 tan p
tan }
}
REVIEW
CONJUGATES
For help with using
conjugates to rationalize
denominators, see
p. 266.
3
4
5}
p
p
1 2 tan } tan }
3
Sum formula for tangent
4
}
Ï3 1 1
5}
}
1 2 Ï3 p 1
}
5 22 2 Ï3
Evaluate.
Simplify.
14.6 Apply Sum and Difference Formulas
n2pe-1406.indd 949
949
10/17/05 11:19:11 AM
EXAMPLE 2
Use a difference formula
4 with p < a < 3p and sin b 5 5
Find cos (a 2 b) given that cos a 5 2}
}
}
5
2
13
p.
with 0 < b < }
2
Solution
3 and
Using a Pythagorean identity and quadrant signs gives sin a 5 2}
5
12 .
cos b 5 }
13
cos (a 2 b) 5 cos a cos b 1 sin a sin b
Difference formula for cosine
4 12 1 2 3 5
5 2}
}
} }
Substitute.
63
5 2}
Simplify.
1 2 1
5 13
5
2 1 13 2
65
✓
GUIDED PRACTICE
for Examples 1 and 2
Find the exact value of the expression.
1. sin 1058
5p
3. tan }
12
2. cos 758
p
4. cos }
12
8 with 0 < a < p and cos b 5 2 24
5. Find sin (a 2 b) given that sin a 5 }
}
}
17
2
25
3
p
with π < b < }.
2
EXAMPLE 3
Simplify an expression
Simplify the expression cos (x 1 p).
cos (x 1 π) 5 cos x cos p 2 sin x sin p
Sum formula for cosine
5 (cos x)(21) 2 (sin x)(0)
Evaluate.
5 2cos x
Simplify.
EXAMPLE 4
Solve a trigonometric equation
p 1 sin x 2 p 5 1 for 0 ≤ x < 2p.
Solve sin 1 x 1 }
2
1 }2
3
3
ANOTHER WAY
You can also solve
by using a graphing
calculator. First graph
each side of the original
equation and then use
the intersect feature
to find the x-value(s)
where the expressions
are equal.
p 1 sin x 2 p 5 1
sin 1 x 1 }
2
1 }2
3
3
p 1 cos x sin p 1 sin x cos p 2 cos x sin p 5 1
sin x cos }
}
}
}
3
3
1
2
}
Ï3
2
3
1
2
3
Write equation.
Use formulas.
}
Ï3
2
} sin x 1 } cos x 1 } sin x 2 } cos x 5 1
Evaluate.
sin x 5 1
Simplify.
p.
c In the interval 0 ≤ x < 2π, the only solution is x 5 }
2
950
n2pe-1406.indd 950
Chapter 14 Trigonometric Graphs, Identities, and Equations
10/17/05 11:19:14 AM
EXAMPLE 5
Solve a multi-step problem
DAYLIGHT HOURS The number h of hours of daylight for Dallas, Texas, and
Anchorage, Alaska, can be approximated by the equations below, where t is the
time in days and t 5 0 represents January 1. On which days of the year will the
two cities have the same amount of daylight?
πt 2 1.35 1 12.1
Dallas: h1 5 2 sin }
182
1
πt 1 12.1
Anchorage: h2 5 26 cos }
182
2
1
2
Solution
STEP 1
Solve the equation h1 5 h2 for t.
πt 2 1.35 1 12.1 5 26 cos πt 1 12.1
2 sin }
}
1 182
2
1 182 2
πt 2 1.35 5 23 cos πt
sin }
}
1 182
2
1 182 2
πt cos 1.35 2 cos πt sin 1.35 5 23 cos πt
sin }
}
}
1 182 2
1 182 2
1 182 2
πt (0.219) 2 cos πt (0.976) 5 23 cos πt
sin }
}
}
1 182 2
1 182 2
1 182 2
πt 5 22.024 cos πt
0.219 sin }
}
1 182 2
1 182 2
πt 5 29.242
tan }
1 182 2
πt
182
21
} 5 tan (29.242) 1 nπ
πt
182
} ø 21.463 1 nπ
t ø 284.76 1 182n
STEP 2 Find the days within one year (365 days) for which Dallas and
Anchorage will have the same amount of daylight.
t ø 284.76 1 182(1) ø 97, or on April 8
t ø 284.76 1 182(2) ø 279, or on October 7
✓
GUIDED PRACTICE
for Examples 3, 4, and 5
Simplify the expression.
6. sin (x 1 2π)
7. cos (x 2 2π)
8. tan (x 2 π)
p t 1 5 5 224 sin p t 1 22 1 5 for 0 ≤ t < 2π.
9. Solve 6 cos }
}
75
75
1 2
1
2
14.6 Apply Sum and Difference Formulas
n2pe-1406.indd 951
951
10/17/05 11:19:15 AM
14.6
EXERCISES
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 9, 23, and 43
★
5 STANDARDIZED TEST PRACTICE
Exs. 2, 11, 18, 32, and 44
SKILL PRACTICE
1. VOCABULARY Give the sum and difference formulas for sine, cosine, and
tangent.
2. ★ WRITING Explain how you can evaluate tan 758 using either the sum or
difference formula for tangent.
EXAMPLE 1
on p. 949
for Exs. 3–10
FINDING VALUES Find the exact value of the expression.
3. tan (2158)
4. sin (21658)
5. tan 1958
23p
7. sin }
12
17p
8. tan }
12
5p
9. cos 2}
12
1
6. cos 158
7p
10. sin 2}
12
2
1
2
p 2 u 5 cos u
11. ★ SHORT RESPONSE Derive the cofunction identity sin 1 }
2
2
using the difference formula for sine.
4
5
EXAMPLE 2
EVALUATING EXPRESSIONS Evaluate the expression given that cos a 5 }
on p. 950
for Exs. 12–18
p and sin b 5 2 15 with 3p < b < 2p.
with 0 < a < }
}
}
2
17
2
12. sin (a 1 b)
13. cos (a 1 b)
14. tan (a 1 b)
15. sin (a 2 b)
16. cos (a 2 b)
17. tan (a 2 b)
3
18. ★ MULTIPLE CHOICE What is the value of sin (a 2 b) given that sin a 5 2}
5
3p and cos b 5 12 with 0 < b < p ?
with π < a < }
}
}
2
13
2
18
A 2}
16
B 2}
55
14
C }
65
20
D }
45
EXAMPLE 3
SIMPLIFYING EXPRESSIONS Simplify the expression.
on p. 950
for Exs. 19–31
19. tan (x 1 π)
43
20. sin (x 1 π)
21. cos (x 1 2π)
22. tan (x 2 2π)
2
p
24. tan 1 x 1 }
22
3p
25. sin x 1 }
2
2
3p
26. cos x 2 }
2
1
2
3p
27. tan x 1 }
2
p
28. cos 1 x 2 }
22
5p
29. tan x 1 }
2
5p
30. cos x 1 }
2
2
3p
23. sin x 2 }
2
1
1
2
1
1
1
2
31. ERROR ANALYSIS Describe and correct the error in simplifying the
expression.
p
tan x 1 tan }
p
4 5 tan x 1 1 5 1
tan 1 x 1 } 2 5 }
}
p
1 1 tan x
4
1 1 tan x tan }
4
EXAMPLE 4
on p. 950
for Exs. 32–38
32. ★ MULTIPLE CHOICE What is a solution of the equation
sin (x 2 2π) 1 tan (x 2 2π) 5 0 on the interval π < x < 3π?
p
A }
2
952
n2pe-1406.indd 952
3p
B }
2
C 2π
D 3π
Chapter 14 Trigonometric Graphs, Identities, and Equations
10/17/05 11:19:15 AM
SOLVING TRIGONOMETRIC EQUATIONS Solve the equation for 0 ≤ x < 2p.
p 2 1 5 cos x 2 p
33. cos 1 x 1 }
1 }6 2
62
p 1 sin x 2 p 5 0
34. sin 1 x 1 }
1 }4 2
42
5p 1 sin x 2 5p 5 1
35. sin x 1 }
}
6
6
p 50
36. tan (x 1 π) 1 cos 1 x 1 }
22
37. tan (x 1 π) 1 2 sin (x 1 π) 5 0
38. sin (x 1 π) 1 cos (x 1 π) 5 0
1
2
1
2
39. CHALLENGE Consider a complex number z 5 a 1 bi in the
imaginary
complex plane shown. Let r be the length of the line segment
joining z and the origin, and let u be the angle that this segment
makes with the positive real axis, as shown.
z 5 a 1 bi
r
a. Explain why a 5 r cos u and b 5 r sin u, so that
z 5 (r cos u) 1 i(r sin u).
b. Use the result from part (a) to show the following:
2
b
u
a
2
real
z 5 r [(cos u cos u 2 sin u sin u) 1 i(sin u cos u 1 cos u sin u)]
c. Use the sum and difference formulas to show that the equation
in part (b) can be written as z2 5 r 2 (cos 2u 1 i sin 2u).
PROBLEM SOLVING
EXAMPLE 5
on p. 951
for Exs. 40–41
40. METEOROLOGY The number h of hours of daylight for Rome, Italy, and
Miami, Florida, can be approximated by the equations below, where t is the
time in days and t 5 0 represents January 1.
p t 2 4.94 1 12.1 Miami: h 5 21.6 cos p t 1 12.1
Rome: h 5 2.7 sin }
}
1 182
1
2
2
182
On which days of the year will the cities have the same amount of daylight?
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
41. CLOCK TOWER The heights m and h (in feet) of a clock
tower’s minute hand and hour hand, respectively, can be
approximated by
pt 2 p
m 5 182.5 2 11.5 sin 1 }
}2
30
2
pt
and h 5 182.5 2 7 sin 1 }
2
360
where t is the time in minutes and t 5 0 represents 3:00 P.M.
Use a graphing calculator to find how long it takes for the
height of the minute hand to equal the height of the hour hand.
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
42. PHYSICAL SCIENCE When a wave travels through a taut string, the
displacement y of each point on the string depends on the time t and the
point’s position x. The equation of a standing wave can be obtained by
adding the displacements of two waves traveling in opposite directions.
Suppose two waves can be modeled by these equations:
2p t 2 2p x
y1 5 A cos }
}
1
3
5
2p t 1 2p x
y 2 5 A cos }
}
2
1
3
5
2
2p t cos 2p x .
Show that y1 1 y 2 5 2A cos }
}
1
3
2
1
5
2
14.6 Apply Sum and Difference Formulas
n2pe-1406.indd 953
953
10/17/05 11:19:17 AM
43. MULTI-STEP PROBLEM A photographer is at a height h taking
camera
aerial photographs. The ratio of the image length WQ to the
length NA of the actual object is
†
f tan (u 2 t) 1 f tan t
WQ
}5 }
NA
h tan u
h
W
Q
t8
where f is the focal length of the camera, u is the angle between
the vertical line perpendicular to the ground and the line from
the camera to point A, and t is the tilt angle of the film.
N
a. Use the difference formula for tangent to simplify the ratio.
A
f
WQ
b. Show that } 5 } when t 5 0.
NA
h
44. ★ EXTENDED RESPONSE Your friend pulls on a weight attached to a spring
and then releases it. A split second later, you begin filming the spring to
analyze its motion. You find that the spring’s distance y (in inches) from its
3
equilibrium point can be modeled by y 5 5 sin (2t 1 C) where C 5 tan21 }
4
and t is the elapsed time (in seconds) since you began filming.
a. Find the values of sin C and cos C.
y
b. Use a sum formula to show that
y 5 5 sin (2t 1 C) can be written
as y 5 4 sin 2t 1 3 cos 2t.
t
c. Graph the function found in
Filming
begins
part (b) and find its maximum
value. Explain what this value
represents.
45. CHALLENGE The busy signal on a touch-tone phone is a combination of two
tones with frequencies of 480 hertz and 620 hertz. The individual tones can
be modeled by the following equations:
480 hertz: y1 5 cos 960πt
620 hertz: y 2 5 cos 1240πt
The sound of the busy signal can be modeled by y1 1 y 2. Show that:
y1 1 y 2 5 2 cos 1100πt cos 140πt
MIXED REVIEW
Sketch the angle. Then find its reference angle. (p. 866)
46. 2308
47. 25108
48. 7308
49. 23558
50. 4608
51. 8608
Solve n ABC. (p. 882)
C
52. A 5 1108, C 5 308, a 5 15
53. B 5 208, b 5 10, c 5 11
54. C 5 1038, B 5 288, b 5 26
55. C 5 1228, a 5 12, c 5 18
PREVIEW
Solve the equation in the interval 0 ≤ x < 2p. (p. 931)
Prepare for
Lesson 14.7
in Exs. 56–61.
56. tan x 1 Ï 3 5 0
954
}
2
59. 5 tan x 2 15 5 0
57. 25 1 8 sin x 5 21
2
60. 4 sin x 2 3 5 0
EXTRA PRACTICE for Lesson 14.6, p. 1023
a
b
A
c
B
58. 4 cos x 5 2 cos x 2 1
61. 2 cos2 x 2 sin x 2 1 5 0
ONLINE QUIZ at classzone.com
14.7
Before
Now
Why?
Key Vocabulary
• sine, p. 852
• cosine, p. 852
• tangent, p. 852
Apply Double-Angle and
Half-Angle Formulas
You evaluated expressions using sum and difference formulas.
You will use double-angle and half-angle formulas.
So you can find the distance an object travels, as in Example 4.
In this lesson, you will use formulas for double angles (angles of measure 2a)
a .
and half angles 1 angles of measure }
2
2
For Your Notebook
KEY CONCEPT
Double-Angle and Half-Angle Formulas
Double-Angle Formulas
cos 2a 5 1 2 2 sin 2 a
sin 2a 5 2 sin a cos a
2 tan a
tan 2a 5 }
2
1 2 tan a
2
cos 2a 5 2 cos a 2 1
cos 2a 5 cos2 a 2 sin 2 a
Half-Angle Formulas
Î
}
2
a 5 6 1 1 cos a
cos }
}
2
a
2
Î
}
a 5 6 1 2 cos a
sin }
}
2
2
a 5 1 2 cos a
tan }
}
2
sin a
a 5 sin a
tan }
}
a
2
The signs of sin } and cos } depend on the
2
1 1 cos a
a
quadrant in which } lies.
2
EXAMPLE 1
Evaluate trigonometric expressions
p.
Find the exact value of (a) cos 1658 and (b) tan }
12
1 (3308)
a. cos 1658 5 cos }
2
CHOOSE SIGNS
Because 1658 is in
Quadrant II and the
value of cosine is
negative in Quadrant II,
the following formula
is used:
Î
}
a
1 1 cos a
cos } 5 2 }
2
2
Î
}}
1 1 cos 3308
52 }
Î
2
}
}
Ï3
11}
2
52 }
2
}
}
Ï2 1 Ï3
5 2}
2
p
1 p
b. tan } 5 tan }
}
12
2162
p
1 2 cos }
6
5}
p
sin }
6
}
Ï3
12}
2
5}
1
2
}
}
5 2 2 Ï3
14.7 Apply Double-Angle and Half-Angle Formulas
n2pe-1407.indd 955
955
10/17/05 11:20:48 AM
EXAMPLE 2
Evaluate trigonometric expressions
5 with 3p < a < 2p, find (a) sin 2a and (b) sin a .
Given cos a 5 }
}
}
13
2
2
Solution
12 .
a. Using a Pythagorean identity gives sin a 5 2}
13
MULTIPLY AN
INEQUALITY
In part (b), you can
multiply through the
3p
inequality } < a < 2π
2
3p
a
1
by } to get } < } < π.
2
4
12 5 5 2 120
sin 2a 5 2 sin a cos a 5 2 2}
}
}
1
a
2
★
21 13 2
169
a is in Quadrant II, sin a is positive.
b. Because }
}
2
2
Î
}
2
So, } is in Quadrant II.
13
a5
sin }
2
Î
5
}
}
12}
1 2 cos a 5
2Ï 13
13
4
}
}5 }5}
2
2
13
13
}
EXAMPLE 3
Î
Standardized Test Practice
sin 2u
Which expression is equivalent to }
?
1 2 cos 2u
B cot u
A sin u
C csc u
D cos u
Solution
2 sin u cos u
1 2 (1 2 2 sin u)
sin 2u
}5 }
2
1 2 cos 2u
Use double-angle formulas.
2 sin u cos u
5}
2 sin2u
Simplify denominator.
cos u
5}
sin u
Divide out common factor 2 sin u.
5 cot u
Use cotangent identity.
c The correct answer is B. A B C D
✓
GUIDED PRACTICE
for Examples 1, 2, and 3
Find the exact value of the expression.
5p
2. sin }
8
p
1. tan }
8
3. cos 158
}
Ï2
p , find cos 2a and tan a .
4. Given sin a 5 }
with 0 < a < }
}
2
2
2
3 with π < a < 3p , find sin 2a and sin a .
5. Given cos a 5 2}
}
}
5
2
2
Simplify the expression.
cos 2u
6. }
sin u 1 cos u
956
n2pe-1407.indd 956
tan 2x
7. }
tan x
x
8. sin 2x tan }
2
Chapter 14 Trigonometric Graphs, Identities, and Equations
10/17/05 11:20:51 AM
PATH OF A PROJECTILE The path traveled by an object that is projected at
an initial height of h0 feet, an initial speed of v feet per second, and an initial
angle θ is given by
16
y52}
x2 1 (tan u)x 1 h0
v 2 cos2 u
where x is the horizontal distance (in feet) and y is the vertical distance (in feet).
(This model neglects air resistance.)
EXAMPLE 4
Derive a trigonometric model
SOCCER Write an equation for the horizontal distance traveled by a soccer ball
kicked from ground level (h0 5 0) at speed v and angle u.
u
Solution
16
USE ZERO PRODUCT
PROPERTY
One solution of this
equation is x 5 0,
which corresponds to
the point where the
ball leaves the ground.
This solution is ignored
in later steps, because
the problem requires
finding where the ball
lands.
2}
x2 1 (tan u)x 1 0 5 0
v 2 cos2 u
Let h0 5 0.
16
2x }
x 2 tan u 5 0
v 2 cos2 u
1
2
Factor.
16
v cos u
x 2 tan u 5 0
}
2
2
Zero product property
16
v cos u
x 5 tan u
}
2
2
Add tan u to each side.
1 v 2 cos2 u tan u
x5}
Multiply each side by }v2 cos2 u.
1 v 2 cos u sin u
x5}
Use cos u tan u 5 sin u.
16
16
1
16
1 v 2 (2 cos u sin u) Rewrite }
1
1
x5}
as } p 2.
32
1 v 2 sin 2u
x5}
32
✓
GUIDED PRACTICE
16
32
Use a double-angle formula.
for Example 4
9. WHAT IF? Suppose you kick a soccer ball from ground level with an initial
speed of 70 feet per second. Can you make the ball travel 200 feet?
1 v 2 sin 2u to explain why the projection
10. REASONING Use the equation x 5 }
32
angle that maximizes the distance a soccer ball travels is u 5 458.
14.7 Apply Double-Angle and Half-Angle Formulas
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957
10/17/05 11:20:52 AM
EXAMPLE 5
Verify a trigonometric identity
Verify the identity cos 3x 5 4 cos3 x 2 3 cos x.
cos 3x 5 cos (2x 1 x)
Rewrite cos 3x as cos (2x 1 x).
5 cos 2x cos x 2 sin 2x sin x
Use a sum formula.
5 (2 cos2 x 2 1) cos x 2 (2 sin x cos x) sin x
Use double-angle formulas.
3
2
5 2 cos x 2 cos x 2 2 sin x cos x
Multiply.
5 2 cos3 x 2 cos x 2 2(1 2 cos2 x) cos x
3
3
5 2 cos x 2 cos x 2 2 cos x 1 2 cos x
3
5 4 cos x 2 3 cos x
EXAMPLE 6
Use a Pythagorean identity.
Distributive property
Combine like terms.
Solve a trigonometric equation
Solve sin 2x 1 2 cos x 5 0 for 0 ≤ x < 2p.
Solution
sin 2x 1 2 cos x 5 0
Write original equation.
2 sin x cos x 1 2 cos x 5 0
Use a double-angle formula.
2 cos x (sin x 1 1) 5 0
Factor.
Set each factor equal to 0 and solve for x.
2 cos x 5 0
sin x 1 1 5 0
cos x 5 0
sin x 5 21
p , 3p
x5}
}
2
3p
x5}
2
2
CHECK Graph the function y 5 sin 2x 1 2 cos x on
a graphing calculator. Then use the zero feature
to find the x-values on the interval 0 ≤ x < 2π for
which y 5 0. The two x-values are:
p ø 1.57
x5}
2
EXAMPLE 7
and
3p ø 4.71
x5}
2
Zero
X=1.570796 Y=0
Find a general solution
x 5 1.
Find the general solution of 2 sin }
2
SOLVE EQUATIONS
As seen in Example 7,
some equations that
involve double or half
angles can be solved
without resorting to
double- or half-angle
formulas.
958
n2pe-1407.indd 958
x 51
2 sin }
Write original equation.
2
x51
sin }
}
2
2
x
2
p
6
Divide each side by 2.
} 5 } 1 2nπ
or
} 1 2nπ
5p
6
General solution for }
p 1 4nπ
x5}
or
} 1 4nπ
5p
3
General solution for x
3
x
2
Chapter 14 Trigonometric Graphs, Identities, and Equations
10/17/05 11:20:52 AM
✓
GUIDED PRACTICE
for Examples 5, 6, and 7
Verify the identity.
11. sin 3x 5 3 sin x 2 4 sin3 x
12. 1 1 cos 10x 5 2 cos2 5x
Solve the equation.
x 1150
14. 2 cos }
2
13. tan 2x 1 tan x 5 0 for 0 ≤ x < 2π
14.7
EXERCISES
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 7, 13, and 53
★
5 STANDARDIZED TEST PRACTICE
Exs. 2, 11, 27, 54, and 55
SKILL PRACTICE
1. VOCABULARY Copy and complete: sin 2a 5 2 sin a cos a is called the ?
formula for sine.
2. ★ WRITING Explain how to determine the sign of the answer when
evaluating a half-angle formula for sine or cosine.
EXAMPLE 1
on p. 955
for Exs. 3–11
EVALUATING EXPRESSIONS Find the exact value of the expression.
3. sin 1058
4. tan 112.58
5. tan (21658)
p
7. cos }
8
5p
8. sin }
12
5p
9. tan 2}
8
1
2
6. cos (2758)
11p
10. sin 2}
12
1
2
11. ★ MULTIPLE CHOICE What is the exact value of tan 158?
}
}
A 2Ï 3
EXAMPLE 2
on p. 956
for Exs. 12–20
B 2 2 Ï3
}
}
C Ï3
D 2 1 Ï3
a
2
a
2
a
2
HALF-ANGLE FORMULAS Find the exact values of sin }, cos }, and tan }.
4, 0 < a < p
12. cos a 5 }
}
5
2
1 , 3p < a < 2π
13. cos a 5 }
}
3 2
12 , p < a < π
14. sin a 5 }
}
13 2
3 , π < a < 3p
15. sin a 5 2}
}
5
2
16. ERROR ANALYSIS Describe and correct the error in finding the exact value of
a given that cos a 5 2 3 with p < a < π.
sin }
}
}
5
2
2
Î
}
Î
}
3
11}
Î5
}
}
a 5 2 1 2 cos a 5 2
2Ï 5
5
4
sin }
}
} 5 2 } 5 2}
2
2
2
5
DOUBLE-ANGLE FORMULAS Find the exact values of sin 2a, cos 2a, and tan 2a.
3p
17. tan a 5 2, π < a < }
2
} p
18. tan a 5 2Ï 3 , }
<a<π
2
2 , π < a < 3p
19. sin a 5 2}
}
3
2
2, 2p < a < 0
20. cos a 5 }
}
5
2
14.7 Apply Double-Angle and Half-Angle Formulas
n2pe-1407.indd 959
959
10/17/05 11:20:53 AM
EXAMPLE 3
SIMPLIFYING EXPRESSIONS Rewrite the expression without double angles or
on p. 956
for Exs. 21–29
p . Then simplify the expression.
half angles, given that 0 < u < }
2
cos 2u
21. }
1 2 2 sin2 u
sin 2u
22. }
2 cos u
cos 2u
24. }
sin u 2 cos u
u
2tan }
2
25. }
csc u
23. (1 2 tan u) tan 2u
u
u
26. 2 sin } cos }
2
2
27. ★ MULTIPLE CHOICE Which expression is equivalent to cot u 1 tan u ?
A csc 2u
B 2 csc 2u
C sec 2u
D 2 sec 2u
ERROR ANALYSIS Describe and correct the error in simplifying the expression.
28.
29.
cos 2x 5 cos2 x 2 sin2 x
}
}
cos2 x
cos2 x
1 (458)
sin 22.58 5 sin }
2
5 2 sin 458 cos 458
1
5}
2
}
cos x
}
Ï2 Ï2
52 }
}
1 2 21 2 2
5 sec2 x
51
EXAMPLE 5
VERIFYING IDENTITIES Verify the identity.
on p. 958
for Exs. 30–35
30. 2 cos2 u 5 1 1 cos 2u
31. sin 3u 5 sin u (4 cos2 u 2 1)
2x 5 sin x cos x
1 sin }
32. }
}
}
3
3
3
2
x 5 2 sin x 2 sin 2x
33. 2 sin 2 x tan }
2
cos 2u
34. 2} 5 2 sin u 2 csc u
sin u
35. cos 4u 5 cos4 u 2 6 sin 2 u cos2 u 1 sin4 u
EXAMPLE 6
on p. 958
for Exs. 36–41
SOLVING EQUATIONS Solve the equation for 0 ≤ x < 2p.
x 51
36. sin }
2
x 1150
37. 2 cos }
2
38. tan x 2 tan 2x 5 0
40. cos 2x 5 22 cos2 x
41. 2 sin 2x sin x 5 3 cos x
}
x 5 2 2 Ï2
39. tan }
}
2
2 sin x
EXAMPLE 7
on p. 958
for Exs. 42–47
FINDING GENERAL SOLUTIONS Find the general solution of the equation.
x 51
42. cos }
2
x 5 sin x
43. tan }
2
44. sin 2x 5 sin x
45. cos 2x 1 cos x 5 0
x 1 sin x 5 0
46. cos }
2
x 1 cos x 5 0
47. sin }
2
48. REASONING Show that the three double-angle formulas for cosine
are equivalent.
49. CHALLENGE Use the diagram shown at the right to derive
u , cos u , and tan u when u is an
the formulas for sin }
}
}
2
2
2
1
u
2
acute angle.
cos u u
1
sin u
960
n2pe-1407.indd 960
5 WORKED-OUT SOLUTIONS
★
5 STANDARDIZED
Chapter 14 Trigonometric
Graphs, Identities, and Equations
on p. WS1
TEST PRACTICE
10/17/05 11:20:55 AM
PROBLEM SOLVING
EXAMPLE 4
on p. 957
for Exs. 50–51
1 v 2 sin 2u from Example 4 on page 957 to find
50. GOLF Use the equation x 5 }
32
the horizontal distance a golf ball will travel if it is hit at an initial speed
of 50 feet per second and at an initial angle of 408.
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
51. SOCCER Suppose you are attempting to kick a soccer ball from ground level.
Through what range of angles can you kick the soccer ball with an initial
speed of 80 feet per second to make it travel at least 150 feet?
"MHFCSB
at classzone.com
52. MULTI-STEP PROBLEM At latitude L, the acceleration
N
due to gravity g (in centimeters per second squared)
at sea level can be approximated by:
45° latitude
30° latitude
g 5 978 1 5.17 sin 2 L 2 0.014 sin L cos L
0° latitude
a. Simplify the equation above to show that
2
g 5 978 1 5.17 sin L 2 0.007 sin 2L.
Equator
b. Graph the function from part (a).
c. Use the graph to approximate the acceleration
S
due to gravity when the latitude is 458, 308, and 08.
53. MACH NUMBER An airplane’s Mach number M is the ratio of its speed to the
speed of sound. When an airplane travels faster than the speed of sound, the
sound waves form a cone behind the airplane. The Mach number is related
to the apex angle u of the cone by the equation sin u 5 1 . Find the angle u
}
2
}
M
that corresponds to a Mach number of 2.5.
54. ★ SHORT RESPONSE A Mercator projection is a map projection of the globe
onto a plane that preserves angles. On a globe with radius r, consider a
point P that has latitude L and longitude T. The coordinates (x, y) of the
corresponding point P9 on the plane can be found using these equations:
x 5 rT
F
1
p
}1L
2
y 5 r ln tan }
2
2G
y
P
P9
x
Œ9
Œ
a. Use half-angle and sum formulas to show that the equation for the
1 1 sin L .
y-coordinate can be written as y 5 r ln }
cos L
1
2
b. What is a reasonable domain for the equation in part (a)? Explain.
14.7 Apply Double-Angle and Half-Angle Formulas
n2pe-1407.indd 961
961
10/17/05 11:20:56 AM
55. ★ EXTENDED RESPONSE At a basketball game, a person has a chance to win
1 million dollars by making a half court shot. The distance from half court to
the point below the 10-foot-high basketball rim is 41.75 feet.
a. Write an equation that models the path of the basketball if the person
releases the ball 6 feet high with an initial speed of 40 feet per second.
b. Simplify the equation. Use a calculator to find the angles at which the
person can make the half court shot.
c. Assume the person releases the ball at one of the angles found in part (b).
What other assumption(s) must you make to say that the shot is made?
56. CHALLENGE A rectangle is inscribed in a
y
semicircle with radius 1, as shown. What value
of u creates the rectangle with the largest area?
(cos u , sin u)
1
u
(cos u , 0)
x
MIXED REVIEW
Graph the function.
57. y 5 3x 2 2 1 1 (p. 123)
58. y 5 2x(x 2 7) (p. 245)
59. f(x) 5 x 2 (x 1 4) (p. 387)
60. y 5 ln x 1 3 (p. 499)
61. y 5 3 tan x (p. 908)
62. y 5 3 tan (x 1 π) (p. 915)
63. 11x 2 2 44x 5 0 (p. 259)
64. z3 1 9z2 5 0 (p. 353)
65. Ï 6x 1 7 5 3 (p. 452)
66. 43x 2 14 5 165x (p. 515)
3 5 7 (p. 589)
67. }
}
x26
x11
68. 2 sin x 2 Ï 3 5 0 (p. 931)
Solve the equation.
}
}
69. SKIING A ski slope at a mountain has an angle of elevation of 27.48. The
vertical height of the slope is 1908 feet. How long is the ski slope? (p. 852)
QUIZ for Lessons 14.6–14.7
Find the exact value of the expression. (pp. 949, 955)
p
1. sin }
2. sin (222.58)
3. tan (23458)
12
p
4. cos }
8
Solve the equation for 0 ≤ x < 2p.
p 2 sin x 2 p 5 0 (p. 949)
5. sin 1 x 1 }
1 }2 2
22
6. cos 2x 5 3 sin x 1 2 (p. 955)
a , cos a , and tan 2a. (p. 955)
Find the exact values of sin }
}
2
2
3, 0 < a < p
7. tan a 5 }
}
5
2
4 , π < a < 3p
8. cos a 5 2}
}
7
2
1 v 2 sin 2u to find the horizontal distance
9. FOOTBALL Use the formula x 5 }
32
x (in feet) that a football travels if it is kicked from ground level with an
initial speed of 25 feet per second at an angle of 308. (p. 955)
962
n2pe-1407.indd 962
PRACTICE
Lesson 14.7,
1023
Chapter 14EXTRA
Trigonometric
Graphs,for
Identities,
andp.Equations
ONLINE QUIZ at classzone.com
10/17/05 11:20:57 AM
MIXED REVIEW of Problem Solving
STATE TEST PRACTICE
classzone.com
Lessons 14.5–14.7
1. MULTI-STEP PROBLEM Suppose two
middle-A tuning forks are struck at different
times so that their vibrations are slightly
out of phase. The combined pressure
change P (in pascals) caused by the tuning
forks at time t (in seconds) is given by:
4. GRIDDED ANSWER In the figure shown below,
the acute angle of intersection, u2 2 u1, of two
lines with slopes m1 and m2 is given by this
equation:
m 2m
1 1 m1m2
2
1
tan (u2 2 u1) 5 }
P 5 3 sin 880πt 1 4 cos 880πt
a. Graph the equation on a graphing
y
m 2 5 tan u2
calculator. Use a viewing window of
0 ≤ t ≤ 0.005 and 26 ≤ P ≤ 6. What do
you observe about the graph?
u2 2 u1
b. Write the given model in the form
u1
m 1 5 tan u1
u2
y 5 a cos b(x 2 h).
x
c. Graph the model from part (b) to confirm
that the graphs are the same.
Find the acute angle of intersection of the
that is a multiple of 308 or 458. Find an angle
that is not a special angle but that can be
written as the sum or difference of two special
angles. Then use a sum or difference formula
to find the cosine of the angle.
3. SHORT RESPONSE The formula for the index
of refraction n of a transparent material is the
ratio of the speed of light in a vacuum to the
speed of light in the material. Some common
materials and their indices are air (1.00), water
(1.33), and glass (1.5). Triangular prisms are
often used to measure the index of refraction
based on the formula shown below.
1 x and y 5 2x 2 4. Round your
lines y 5 }
2
answer to the nearest tenth of a degree.
5. EXTENDED RESPONSE The graph below shows
the average daily temperature T (in degrees
Fahrenheit) in Denver, Colorado. The time t is
measured in months, with t 5 0 representing
January 1.
Daily Temperature in Denver
Temperature
(8F)
2. OPEN-ENDED Recall that a special angle is one
T
80
(6, 73)
40
(0, 29)
0
a
u
light
0
2
4
6
8
10
Months since January 1
t
a. Tell whether the graph
u
a
sin } 1 }
12
2
should be modeled by a
sine or cosine function.
Explain your reasoning.
2
n5}
u
sin }
2
b. Write a trigonometric
a. Show that the index of refraction can be
simplified as shown:
a 1 sin a cot u
n 5 cos }
}
}
2
2
2
b. Use a graphing calculator to find the value
of u when a 5 608 and the prism is made of
glass.
model for the average
daily temperature in
Denver, Colorado.
c. On which days of the
year is the average daily
temperature in Denver,
Colorado, 708F?
Mixed Review of Problem Solving
n2pe-1407.indd 963
963
10/17/05 11:21:00 AM
14
CHAPTER SUMMARY
Big Idea 1
Algebra
classzone.com
Electronic Function Library
BIG IDEAS
For Your Notebook
Graphing Trigonometric Functions
The graphs of y 5 a sin bx and y 5 a cos bx are shown below for a > 0 and b > 0.
y 1 2π
y
s 4 ? b , ad
y 5 a sin bx
y 5 a cos bx
(0, a)
s2πb , ad
s 14 ? 2πb , 0d
s 12 ? 2πb , 0d s2πb , 0d
x
(0, 0)
s 34 ? 2πb , 0d
x
s 12 ? 2πb , 2ad
s 34 ? 2πb , 2ad
To graph a function of the form y 5 a sin b(x 2 h) 1 k or y 5 a cos b(x 2 h) 1 k,
shift the graph of y 5 a sin bx or y 5 a cos bx, respectively, horizontally h units
and vertically k units. Then, if a < 0, reflect the graph in the midline y 5 k.
Big Idea 2
Solving Trigonometric Equations
The table below shows strategies that may help you solve trigonometric
equations. Only the first few steps are shown.
Factor
Use the quadratic formula
x sin2 x 2 x 5 0
x (sin2 x 2 1) 5 0
cos2 x 2 6 cos x 1 1 5 0
}}
6 6 Ï 36 2 4(1)(1)
2(1)
cos x 5 }
x (sin x 1 1)(sin x 2 1) 5 0
}
Use an identity
cos2 x 2 1 5 9 sin2 x
cos2 x 2 1 5 9(1 2 cos2 x)
10 cos2 x 5 10
cos x 5 3 6 2Ï 2
Big Idea 3
cos2 x 5 1
Applying Trigonometric Formulas
Use the formulas below to evaluate trigonometric functions of certain angles.
Sum
formulas
sin (a 1 b) 5 sin a cos b 1 cos a sin b
cos (a 1 b) 5 cos a cos b 2 sin a sin b
tan (a 1 b) 5 }
Difference
formulas
sin (a 2 b) 5 sin a cos b 2 cos a sin b
cos (a 2 b) 5 cos a cos b 1 sin a sin b
tan (a 2 b) 5 }
cos 2a 5 cos2 a 2 sin2 a
Double-angle
formulas
2
cos 2a 5 2 cos a 2 1
cos 2a 5 1 2 2 sin2 a
sin 2a 5 2 sin a cos a
Î 2
2
1 1 cos a
a
cos 5 6Î
2
2
}
Half-angle
formulas
1 2 cos a
a
sin } 5 6 }
}
}
964
n2pe-1480.indd 964
}
tan a 1 tan b
1 2 tan a tan b
tan a 2 tan b
1 1 tan a tan b
2 tan a
1 2 tan a
tan 2a 5 }
2
a
2
1 2 cos a
sin a
a
2
sin a
1 1 cos a
tan } 5 }
tan } 5 }
Chapter 14 Trigonometric Graphs, Identities, and Equations
10/17/05 11:22:42 AM
14
CHAPTER REVIEW
classzone.com
• Multi-Language Glossary
• Vocabulary practice
REVIEW KEY VOCABULARY
• amplitude, p. 908
• period, p. 908
• trigonometric identity, p. 924
• periodic function, p. 908
• frequency, p. 910
• sinusoid, p. 941
• cycle, p. 908
VOCABULARY EXERCISES
1. Copy and complete: Frequency gives the number of ? per unit of time.
2. WRITING Explain how to find the period of y 5 a sin b(x 2 h) 1 k.
Determine whether the given number is the amplitude, period, or frequency of
px .
the graph of y 5 p cos }
2
4. π
3. 4
5. 0.25
REVIEW EXAMPLES AND EXERCISES
Use the review examples and exercises below to check your understanding of the
concepts you have learned in each lesson of Chapter 14.
14.1
Graph Sine, Cosine, and Tangent Functions
pp. 908–914
EXAMPLE
1 cos 2x and (b) y 5 3 tan x .
Graph (a) y 5 }
}
2
2
2p 5 π
Period: }
1
a. Amplitude: a 5 }
2
y
2
1
2
p , 0 ; (p, 0)
Intercepts: (0, 0); 1 }
2
2
3p
1
Minimum: }
, 2}
p, 1
Maximum: 1 }
}2
1
4 2
p 5 2π
b. Period: }
4
2
x
π
4
2
Intercept: (0, 0)
1
2
y
}
1
Asymptotes: x 5 2p ; x 5 p
2π
π
8. g(x) 5 5 sin πx
1x
9. y 5 2 tan }
3
x
p , 23 ; p , 3
Halfway points: 1 2}
2 1} 2
2
EXAMPLES
1, 2, and 4
on pp. 909–912
for Exs. 6–9
2
EXERCISES
Graph the function.
6. y 5 sin 2x
1 cos x
7. f(x) 5 }
}
2
2
Chapter Review
n2pe-1480.indd 965
965
10/17/05 11:22:46 AM
14
14.2
CHAPTER REVIEW
Translate and Reflect Trigonometric Graphs
pp. 915–922
EXAMPLE
Graph y 5 3 cos (x 2 p) 2 1.
To graph y 5 3 cos (x 2 π) 2 1, start with the graph of y 5 3 cos x. Then, translate
the graph right π units and down 1 unit.
Amplitude: 3 5 3
Period: 2π
Horizontal shift: π
Vertical shift: 21
3p
5p
On y 5 k: }
, 21 ; }
, 21
1
2
21
2
y
1
π
2
x
Minimum: (2p, 24)
Maximums: (p, 2); (3p, 2)
EXERCISES
EXAMPLES
1, 2, and 4
Graph the function.
on pp. 915–917
for Exs. 10–15
10. f(x) 5 cos 2x 1 4
1 sin 5(x 2 π)
11. y 5 }
2
p 13
12. y 5 2 sin 1 x 2 }
22
1x 1 3
13. y 5 2 cos }
3
14. g(x) 5 21 2 3 cos 4x
p
15. y 5 4 2 sin 3 1 x 2 }
32
14.3
Verify Trigonometric Identities
pp. 924–930
EXAMPLE
cot2 u
Verify the identity }
5 csc u 2 sin u.
csc u
cot2 u
csc u
csc2 u 2 1
csc u
}5}
Pythagorean identity
csc2 u
1
5}
2}
csc u
csc u
Write as separate fractions.
1
5 csc u 2 }
csc u
Simplify.
5 csc u 2 sin u
Reciprocal identity
EXERCISES
EXAMPLES
2, 3, 4, and 5
Simplify the expression.
on pp. 925–926
for Exs. 16–20
16. 2cos x tan (2x)
17. sec x tan 2 x 1 sec x
p 2 x tan x
18. sin 1 }
2
2
Verify the identity.
sin2 (2x) 2 1
19. }
5 2sin 2 x
cot2 x
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p 2 x cot x 5 csc2 x 2 1
20. tan 1 }
2
2
Chapter 14 Trigonometric Graphs, Identities, and Equations
10/17/05 11:22:48 AM
classzone.com
Chapter Review Practice
14.4
Solve Trigonometric Equations
pp. 931–937
EXAMPLE
Solve 2 cos2 x 5 1 in the interval 0 ≤ x < 2p.
2 cos2 x 5 1
Write original equation.
1
cos2 x 5 }
Divide each side by 2.
2
}
Ï2
cos x 5 6}
Take square roots of each side.
2
p , 3p , 5p , and 7p .
c In the interval 0 ≤ x < 2π, the solutions are x 5 }
} }
}
4
EXAMPLES
1 and 4
on pp. 931–933
for Exs. 21–23
14.5
4
4
4
EXERCISES
Solve the equation in the interval 0 ≤ x < 2p.
21. 24 sin 2 x 5 23
22. cos2 x 5 cos x
23. tan 2 4x 5 3
Write Trigonometric Functions and Models
pp. 941–947
EXAMPLE
Write a function for the sinusoid.
STEP 1
y
Find the maximum value M and minimum
value m. From the graph, M 5 3 and m 5 21.
sπ2 , 3d
3
STEP 2 Identify the vertical shift, k.
2
M 1 m 5 3 1 (21) 5 }
k5}
51
}
2
2
2
s
π
,
6
x
π
2
d
21
STEP 3 Decide whether the graph should be modeled by a sine or cosine function.
Because the graph crosses the midline, y 5 1, on the y-axis and then
decreases to its minimum value, the graph is a sine curve with a reflection
but no horizontal shift. So, a < 0 and h 5 0.
2p 5 2p . So, b 5 3.
STEP 4 Find the amplitude and period. The period is }
}
3
b
3 2 (21)
M
2
m
4
The amplitude is a 5 } 5 } 5 } 5 2. So, a 5 22.
2
2
2
A function for the sinusoid is y 5 2 2 sin 3x 1 1.
EXERCISES
EXAMPLE 1
on p. 941
for Exs. 24–25
Write a function for the sinusoid.
24.
y
2
π
,
2
25.
s d
1
y
(0, 21)
1
x
x
π
3
sπ6 , 21d
24
(1, 23)
Chapter Review
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14
14.6
CHAPTER REVIEW
Apply Sum and Difference Formulas
pp. 949–954
EXAMPLE
Find the exact value of the expression.
a. cos 2258 5 cos (2708 2 458)
Substitute 270 8 2 45 8 for 225 8.
5 cos 2708 cos 458 1 sin 2708 sin 458
}
Difference formula for cosine
}
Ï2
Ï2
50 }
21 }
122
122
Evaluate.
}
Ï2
5 2}
Simplify.
2
7p
p
p
b. sin } 5 sin 1 } 1 } 2
4
12
3
p cos p 1 cos p sin p
5 sin }
}
}
}
4
3
}
}
122
}
p
3
4
7p
12
Sum formula for sine
4
3
}
Ï3 Ï2
1 Ï2
5}
} 1} }
2
p
Substitute } 1 } for }.
2
122
Evaluate.
}
Ï6 1 Ï2
5}
Simplify.
4
EXERCISES
EXAMPLES
1 and 2
Find the exact value of the expression.
on pp. 949–950
for Exs. 26–30
26. cos 1958
14.7
13p
28. sin }
12
8
p < a < π and cos b 5 1
30. Find cos (a 2 b), given that sin a 5 } with }
}
17
2
2
p
with 0 < b < }.
2
27. tan 758
Apply Double-Angle and Half-Angle Formulas
5p
29. cos }
6
pp. 955–962
EXAMPLE
Find the exact value of the expression.
1 (2708) 5 1 2 cos 2708 5 1 2 0 5 21
a. tan 1358 5 tan }
}
}
2
21
sin 2708
Î
}
p
1 p 5
b. sin } 5 sin }
}
12
216 2
p
1 2 cos }
6
}5
2
Î
}
}
Ï3
12}
2
1
2
}
}
} 5 }Ï 2 2 Ï 3
2
EXERCISES
EXAMPLES
1 and 2
Find the exact value of the expression.
on pp. 955–956
for Exs. 31–35
31. sin 758
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p
33. cos }
12
1 with 0 < a < p , find sin 2a and tan a .
35. Given cos a 5 }
}
}
2
2
2
32. tan (2158)
3p
34. cos }
4
Chapter 14 Trigonometric Graphs, Identities, and Equations
10/17/05 11:22:51 AM
14
CHAPTER TEST
Graph the function.
1. f(x) 5 4 cos 2x
3 sin π x
2. y 5 }
2
px
3. f(x) 5 24 tan }
2
4. y 5 sin (x 2 π) 2 2
p
5. f(x) 5 3 tan 1 x 2 }
22
1x 1 3
6. y 5 22 cos }
3
Simplify the expression.
sin (2u)
7. }
tan (2u)
2
2
p 2x
sin 1 }
2
2
2
8. cos x 1 sin x 2 csc x
9. }
sec x
Write a function for the sinusoid.
10.
11.
y
(2, 5)
y
2
s 9π8 , 1d
π
π
8
x
1
2
(1, 21)
x
s 3π8 , 25d
12. Verify the identity cos 3x 5 4 cos3 x 2 3 cos x.
Solve the equation in the interval 0 ≤ x < 2p.
}
13. 9 sin 2 x tan x 5 16 tan x
14. (1 2 tan 2 x) tan 2x 5 2Ï 3
}
x 5 Ï2
15. sin }
}
2
2
Find the general solution of the equation.
16. 6 tan 2 x 2 2 5 0
x 5 1 2 cos x
18. sin }
2
17. cos x 5 sin 2x sin x
Find the exact value of the expression.
5p
21. tan }
12
p
20. cos 1 2}
82
19. sin 2558
10p
22. sin }
3
23. BOATING The paddle wheel of a ship is 11 feet in diameter, revolves 15
times per minute when moving at top speed, and is 2 feet below the water’s
surface at its lowest point. Using this speed and starting from a point at
the very top of the wheel, write a model for the height h (in feet) of the
end of the paddle relative to the water’s surface as a function of time t (in
minutes).
24. PRECIPITATION The table below shows the monthly precipitation P
(in inches) in Bismarck, North Dakota. The time t is measured in months,
with t 5 1 representing January. Use a graphing calculator to write a
sinusoidal model that gives P as a function of t.
t
1
2
3
4
5
6
7
8
9
10
11
12
P
0.5
0.5
0.9
1.5
2.2
2.6
2.6
2.2
1.6
1.3
0.7
0.4
Chapter Test
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14
★ Standardized TEST PREPARATION
Scoring Rubric
EXTENDED RESPONSE QUESTIONS
Full Credit
• solution is complete
and correct
Partial Credit
• solution is complete
but has errors,
or
• solution is without
error but incomplete
No Credit
• no solution is given,
or
• solution makes no
sense
PROBLEM
The horizontal distance x (in feet) traveled by a football kicked from
1 v 2 sin 2u.
ground level at speed v and angle u can be modeled by x 5 }}
32
a. Write an expression for half of the horizontal distance. How is this
expression related to the football’s maximum height?
b. Write a simplified expression for the football’s maximum height.
c. What is the maximum height of a football kicked from ground level
with a speed of 80 feet per second and an angle of 358?
Below are sample solutions to the problem. Read each solution and the
comments in blue to see why the sample represents full credit, partial credit,
or no credit.
SAMPLE 1: Full credit solution
The expression
is correct, and its
significance is fully
explained.
1 p 1 v 2 sin 2u 5 1 v 2 sin 2u
a. Half of the horizontal distance 5 }
}}
}}
2 32
64
Because the ball travels in a parabolic path, the maximum height occurs
at half the horizontal distance.
b. Substitute the expression from part (a) for x in the equation for the path of
a projectile. Note that the initial height h0 is 0.
The correct formula is
used for the path of a
projectile.
16
x2 1 (tan u)x 1 h0
y 5 2}}}}
v 2 cos2 u
1
16
}}v 2 sin 2u
y 5 2}}}}
v 2 cos2 u 64
1
2
2
1 v 2 sin 2u 1 0
1 (tan u) }}
1 64
2
v 2 (sin 2u)(sin 2u)
(v 2 sin 2u)tan u
1 }}}}}}}
5 2 }}}}}}}}
2
64
256 cos u
A double-angle formula
is used correctly, and the
resulting expression is
correctly simplified.
v 2(2 sin u cos u)(2 sin u cos u)
v 2(2 sin u)(cos u tan u)
5 2 }}}}}}}}}}}}} 1 }}}}}}}}}}
256 cos u cos u
64
1
1
v2 (2 sin u cos u)(2 sin u cos u)
256 cos u cos u
v 2 (2 sin u)(sin u)
64
5 2 }}}}}}}}}}}}} 1 }}}}}}}}
64
v2 sin2 u
2v 2 sin2 u
5 2 }}}} 1 }}}}}
64
64
1 v 2 sin 2 u
5 }}
64
c. To find the maximum height of the ball, substitute 80 for v and 358 for u.
1 v 2 sin 2 u 5 1 p 802 sin 2 358 ø 33 feet
Maximum height 5 }}
}}
The answer is correct.
970
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64
64
Chapter 14 Trigonometric Graphs, Identities, and Equations
10/17/05 11:23:50 AM
SAMPLE 2: Partial credit solution
The answer is correct, but
work is not shown and
no explanation is given.
1 v 2 sin 2u
a. }}
64
16
b. y 5 2}}}}
x2 1 (tan u)x 1 h0
v 2 cos2 u
1
16
}}v 2 sin 2u
5 2}}}}
v 2 cos2 u 64
1
2
2
1 v 2 sin 2u 1 0
1 (tan u) }}
1 64
2
16
1 4
1 2
2
5 2}}}}
}}}v sin 2u 1 (tan u) }}v sin 2u
64
v 2 cos2 u 4096
1
The expression is not
fully simplified.
The answer is correct.
2
1
2
2
v 2 sin2 2u
tan u sin 2u
1 v}}}}}}}
5 2 }}}}}
2
64
256 cos u
c. The maximum height of the football is:
2
v 2 sin2 2u
tan u sin 2u
802 sin2 708 1 802 tan 358 sin 708 ø 33 feet
2 }}}}}
1 v}}}}}}}
5 2 }}}}}}
}}}}}}}}
2
64
64
256 cos u
256 cos2 358
SAMPLE 3: No credit solution
The expressions in parts
(a) and (b) are incorrect,
and no work is shown.
The values are
substituted incorrectly.
The answer is wrong.
1 v 2 sin u
a. x 5 }}
64
1 v 2 sin u.
b. The maximum height is given by }}
64
1 v 2 sin u 5 1 p 352 sin 808 ø 19 feet
c. x 5 }}
}}
64
64
The maximum height is about 19 feet.
PRACTICE
Apply the Scoring Rubric
Score the following solution to the problem on the previous page as full credit,
partial credit, or no credit. Explain your reasoning. If you choose partial credit
or no credit, explain how you would change the solution so that it earns a score
of full credit.
1 v2 sin 2u. This is the horizontal
a. Half of the total horizontal distance is }}
64
distance when the ball is at its maximum height.
1 v2 sin2 u.
b. A simplified expression for the maximum height of the ball is }}
64
c. Substitute 80 for v and 358 for u in the expression from part (b) to find
the maximum height of the football.
1
64
1
64
1
64
2
2
2
2
2
}}v sin u 5 }} p 80 sin 358 ø }} p 6400(0.574) ø 33 feet
Standardized Test Preparation
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14
★ Standardized TEST PRACTICE
EXTENDED RESPONSE
1. Sound travels in waves that can be represented using sine functions. When
two notes are played at the same time on a musical instrument, the resulting
sound wave can be modeled by the sum of two sound waves.
a. Playing C and G together creates harmony and
produces a pleasant sound. Use a graphing
calculator to graph the sound wave that results
from playing C and G at the same time.
Note
Sound wave
C
y 5 sin 523.26t
G
y 5 sin 784t
B
y 5 sin 987.76t
b. Playing C and B together creates dissonance and
produces an unpleasant sound. Use a graphing
calculator to graph the sound wave that results
from playing B and C at the same time.
c. Make a conjecture about the relationship between how the notes
sound when played together and the resulting sound wave.
Explain your reasoning.
2. The blades of a fan rotate counterclockwise at a
speed of 800 rotations per minute. The length of
each blade is 9 inches. The maximum height of the
tip of the red fan blade above the ground is
48 inches.
a. What is the minimum height of the tip of the red
fan blade above the ground?
b. Write a function that models the height of the tip
of the red fan blade as a function of time. Assume
that the blade starts in the position shown in the
diagram.
c. Do your answers to parts (a) and (b) change if the
fan rotates clockwise? Explain why or why not.
3. The chart shows the average monthly temperature (in degrees Fahrenheit)
and a household’s gas usage (in cubic feet) for 12 months.
a. Use the chart to make a table of
values giving the month t (with
January corresponding to t 5 0),
the average monthly temperature y1,
and the gas usage y 2 (in thousands
of cubic feet).
b. Use the table from part (a) and
a graphing calculator to find
trigonometric models for the average
monthly temperature y1 as a function
of time and the gas usage y 2 as a
function of time.
c. Graph the two regression equations
January
February
March
April
328F
218F
158F
228F
20,000 ft3
27,000 ft3
23,000 ft3
22,000 ft3
May
June
July
August
498F
358F
21,000 ft
3
14,000 ft
628F
3
8,000 ft
788F
3
9,000 ft3
September
October
November
December
718F
638F
558F
408F
13,000 ft3
15,000 ft3
19,000 ft3
23,000 ft3
in the same coordinate plane on your
graphing calculator. Describe the
relationship between the graphs.
972
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STATE TEST PRACTICE
classzone.com
MULTIPLE CHOICE
GRIDDED ANSWER
4. The graph of which function is shown?
y
7. What is the x-intercept of the graph of
1 πx on the interval 0 < x < 3?
y 5 sin }
2
(2π, 3)
3
8. Find the value of tan (a 1 b) given that the
following are true:
x
2π
3 with 0 < a < p and
sin a 5 }
}
(0, 23)
5
2
}
A y 5 23 cos 2x
B y 5 22 cos 3x
1 cos 3x
C y 5 2}
1x
D y 5 23 cos }
2
Ï2
p
cos b 5 }}
with 0 < b < }
2
2
9. Find the y-coordinate of the point of
2
intersection of the graphs of y 5 2 1 sin x
p.
and y 5 3 2 sin x in the interval 0 < x < }
5. Which expression is not equivalent to 1?
A tan x sec x cos x
B sin 2 x 1 cos2 x
cos2 (2x) tan2 x
C }}}}}}}
sin2 (2x)
p 2 x csc x
D cos 1 }
2
2
2
10. Find the amplitude of the sinusoid shown
below.
y
(3π, 4)
4
6. What is the general solution of the equation
cos3 x 5 25 cos x?
p 1 2nπ
A }
p 1 nπ
B }
3p 1 2nπ
C }}
2
p 1 2nπ
D }
4
2
π
2
x
(π, 22)
SHORT RESPONSE
11. Use the fact that the frequency of a periodic function’s graph is the reciprocal
of the period to show that an oscillating motion with maximum displacement
a and frequency f can be modeled by y 5 a sin 2πft or y 5 a cos 2πft.
12. A basketball is dropped from a height of 15 feet. Can the height of the
basketball over time be modeled by a trigonometric function? If so, write the
function. If not, explain why not.
13. Consider the following function:
f(x) 5 sin x cos x
Without graphing the function, what would you expect the graph to look like?
Explain your reasoning. Copy and complete the table below and determine
whether your prediction was correct.
x
2p
3p
2}}
p
2}
p
2}
0
}
p
4
}
p
2
}}
3p
4
p
f(x)
?
?
?
?
?
?
?
?
?
4
2
4
Standardized Test Practice
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