Download Evaluate Trigonometric Functions of Any Angle

13.3
Before
Now
Why?
Key Vocabulary
• unit circle
• quadrantal angle
• reference angle
Evaluate Trigonometric
Functions of Any Angle
You evaluated trigonometric functions of an acute angle.
You will evaluate trigonometric functions of any angle.
So you can calculate distances involving rotating objects, as in Ex. 37.
You can generalize the right-triangle definitions of trigonometric functions from
Lesson 13.1 so that they apply to any angle in standard position.
For Your Notebook
KEY CONCEPT
General Definitions of Trigonometric Functions
Let u be an angle in standard position, and let (x, y)
be the point where the terminal side of u intersects
the circle x 2 1 y 2 5 r 2. The six trigonometric
functions of u are defined as follows:
y
u
(x, y)
r
y
r
sin u 5 }
r, y Þ 0
csc u 5 }
x
cos u 5 }
r, x Þ 0
sec u 5 }
y
x
x, y Þ 0
cot u 5 }
x
y
r
x
tan u 5 }, x Þ 0
y
These functions are sometimes called circular functions.
EXAMPLE 1
Evaluate trigonometric functions given a point
Let (24, 3) be a point on the terminal side of an
angle u in standard position. Evaluate the six
trigonometric functions of u.
y
u
(24, 3)
r
Solution
x
Use the Pythagorean theorem to find the value of r.
}
}
}
r 5 Ïx 2 1 y 2 5 Ï(24)2 1 32 5 Ï 25 5 5
Using x 5 24, y 5 3, and r 5 5, you can write
the following:
y
r
x 5 24
cos u 5 }
}
3
tan u 5 } 5 2}
r 55
csc u 5 }
}
r 5 25
sec u 5 }
}
x 5 24
cot u 5 }
}
y
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y
x
3
sin u 5 } 5 }
5
3
r
x
5
4
y
4
3
Chapter 13 Trigonometric Ratios and Functions
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For Your Notebook
KEY CONCEPT
The Unit Circle
y
The circle x2 1 y 2 5 1, which has center (0, 0)
and radius 1, is called the unit circle. The values
of sin u and cos u are simply the y-coordinate and
x-coordinate, respectively, of the point where the
terminal side of u intersects the unit circle.
y
r
y
1
u
x
r51
(x, y)
x 5 x 5x
cos u 5 }
}
sin u 5 } 5 } 5 y
r
1
It is convenient to use the unit circle to find trigonomet ric functions of
quadrantal angles. A quadrantal angle is an angle in standard position whose
terminal side lies on an axis. The measu re of a quadrantal angle is always a
p radians.
multiple of 908, or }
2
EXAMPLE 2
Use the unit circle
Use the unit circle to evaluate the six trigonometric functions of u 5 2708.
Solution
ANOTHER WAY
The general circle
x2 1 y 2 5 r 2 can also
be used to find the
trigonometric functions
of u 5 2708. The
terminal side of u
intersects the circle at
(0, 2r). Therefore:
y
r
2r
r
sin u 5 } 5 } 5 21
The other functions can
be evaluated similarly.
Draw the unit circle, then draw the angle u 5 2708 in
standard position. The terminal side of u intersects
the unit circle at (0, 21), so use x 5 0 and y 5 21 to
evaluate the trigonometric functions.
u
x
y
21 5 21
sin u 5 } 5 }
r
1
r 5 1 5 21
csc u 5 }
}
y
21
x 5 0 50
cos u 5 }
}
r
1
r 5 1 undefined
sec u 5 }
}
x
0
y
x
(0, 21)
x 5 0 50
cot u 5 }
}
21 undefined
tan u 5 } 5 }
0
"MHFCSB
✓
y
y
21
at classzone.com
GUIDED PRACTICE
for Examples 1 and 2
Evaluate the six trigonometric functions of u.
1.
2.
y
3.
y
(28, 15)
u
u
u
x
(3, 23)
y
x
x
(25, 212)
4. Use the unit circle to evaluate the six trigonometric functions of u 5 1808.
13.3 Evaluate Trigonometric Functions of Any Angle
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For Your Notebook
KEY CONCEPT
Reference Angle Relationships
READING
The symbol u9 is read
as “theta prime.”
Let u be an angle in standard position. The reference angle for u is the acute
angle u9 formed by the terminal side of u and the x-axis. The relationship
between u and u9 is shown below for nonquadrantal angles u such that
p
908 < u < 3608 1 } < u < 2p 2.
2
Quadrant II
Quadrant III
y
u
x
Degrees: u9 5 1808 2 u
Radians: u9 5 p 2 u
EXAMPLE 3
y
u
u
u9
Quadrant IV
y
x
u9
Degrees: u9 5 u 2 1808
Radians: u9 5 u 2 p
u9
x
Degrees: u9 5 3608 2 u
Radians: u9 5 2p 2 u
Find reference angles
5p
Find the reference angle u9 for (a) u 5 }
and (b) u 5 21308.
3
Solution
5p 5 p .
a. The terminal side of u lies in Quadrant IV. So, u9 5 2π 2 }
}
3
3
b. Note that u is coterminal with 2308, whose terminal side lies in
Quadrant III. So, u9 5 2308 2 1808 5 508.
EVALUATING TRIGONOMETRIC FUNCTIONS Reference angles allow you to
evaluate a trigonometric function for any angle u. The sign of the trigonometric
function value depends on the quadrant in which u lies.
KEY CONCEPT
For Your Notebook
Evaluating Trigonometric Functions
Use these steps to evaluate a
trigonometric function for any angle u :
STEP 1
Find the reference angle u9.
STEP 2 Evaluate the trigonometric
function for u9.
STEP 3 Determine the sign of the
trigonometric function value
from the quadrant in which
u lies.
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Signs of Function Values
Quadrant II
sin u , csc u : 1
cos u , sec u : 2
tan u , cot u : 2
Quadrant III
sin u , csc u : 2
cos u , sec u : 2
tan u , cot u : 1
y
Quadrant I
sin u, csc u : 1
cos u, sec u : 1
tan u, cot u : 1
x
Quadrant IV
sin u, csc u : 2
cos u, sec u : 1
tan u, cot u : 2
Chapter 13 Trigonometric Ratios and Functions
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EXAMPLE 4
Use reference angles to evaluate functions
17p
Evaluate (a) tan (22408) and (b) csc }
.
6
Solution
a. The angle 22408 is coterminal with 1208. The
y
reference angle is u9 5 1808 2 1208 5 608. The
tangent function is negative in Quadrant II,
so you can write:
u 9 5 608
x
}
tan (22408) 5 2tan 60 o 5 2Ï 3
u 5 22408
17p is coterminal with 5p . The
b. The angle }
}
6
6
5p 5 p . The
reference angle is u9 5 π 2 }
}
6
6
y
u9 5
cosecant function is positive in Quadrant II,
so you can write:
π
6
x
u5
17p 5 csc p 5 2
csc }
}
6
6
✓
GUIDED PRACTICE
17π
6
for Examples 3 and 4
Sketch the angle. Then find its reference angle.
5. 2108
6. 22608
7p
7. 2}
9
15p
8. }
4
9. Evaluate cos (22108) without using a calculator.
EXAMPLE 5
Calculate horizontal distance traveled
ROBOTICS The “frogbot” is a robot designed for exploring
rough terrain on other planets. It can jump at a 458 angle
and with an initial speed of 16 feet per second. On
Earth, the horizontal distance d (in feet) traveled
by a projectile launched at an angle u and with
an initial speed v (in feet per second) is given by:
Frogbot
v 2 sin 2u
d5}
32
INTERPRET MODELS
This model neglects air
resistance and assumes
that the projectile’s
starting and ending
heights are the same.
How far can the frogbot jump on Earth?
Solution
2
v sin 2u
d5}
32
2
Write model for horizontal distance.
16 sin (2 p 458)
5}
Substitute 16 for v and 45 8 for u.
58
Simplify.
32
c The frogbot can jump a horizontal distance of 8 feet on Earth.
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EXAMPLE 6
Model with a trigonometric function
y
ROCK CLIMBING A rock climber is using a rock
climbing treadmill that is 10.5 feet long. The climber
begins by lying horizontally on the treadmill, which
is then rotated about its midpoint by 1108 so that
the rock climber is climbing towards the top. If the
midpoint of the treadmill is 6 feet above the ground,
how high above the ground is the top of the treadmill?
Solution
5.25 ft
110°
x
?
6 ft
y
r
Use definition of sine.
y
5.25
Substitute 110 8 for u and } 5 5.25 for r.
sin u 5 }
10.5
2
sin 1108 5 }
4.9 ø y
Solve for y.
c The top of the treadmill is about 6 1 4.9 5 10.9 feet above the ground.
✓
GUIDED PRACTICE
for Examples 5 and 6
10. TRACK AND FIELD Estimate the horizontal distance traveled by a track and
field long jumper who jumps at an angle of 208 and with an initial speed of
27 feet per second.
11. WHAT IF? In Example 6, how high is the top of the rock climbing treadmill
if it is rotated 1008 about its midpoint?
13.3
EXERCISES
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 5, 17, and 37
★
5 STANDARDIZED TEST PRACTICE
Exs. 2, 11, 33, 37, and 39
SKILL PRACTICE
1. VOCABULARY Copy and complete: A(n) ? is an angle in standard position
whose terminal side lies on an axis.
2. ★ WRITING Given an angle u in Quadrant III, explain how you can use a
reference angle to find cos u.
EXAMPLE 1
on p. 866
for Exs. 3–11
USING A POINT Use the given point on the terminal side of an angle u in
standard position to evaluate the six trigonometric functions of u.
3. (8, 15)
4. (29, 12)
5. (27, 224)
6. (5, 212)
7. (2, 22)
8. (26, 9)
9. (23, 25)
10. (5, 2Ï 11 )
}
11. ★ MULTIPLE CHOICE Let (27, 24) be a point on the terminal side of an angle u
in standard position. What is the value of tan u ?
7
A 2}
4
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4
B 2}
7
4
C }
7
7
D }
4
Chapter 13 Trigonometric Ratios and Functions
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EXAMPLE 2
QUADRANTAL ANGLES Evaluate the six trigonometric functions of u.
on p. 867
for Exs. 12–15
12. u 5 08
EXAMPLE 3
FINDING REFERENCE ANGLES Sketch the angle. Then find its reference angle.
on p. 868
for Exs. 16–23
16. 21008
17. 1508
18. 3208
19. 23708
5p
20. 2}
6
8p
21. }
3
15p
22. }
4
13p
23. 2}
6
p
13. u 5 }
2
7p
15. u 5 }
2
14. u 5 5408
EXAMPLE 4
EVALUATING FUNCTIONS Evaluate the function without using a calculator.
on p. 869
for Exs. 24–31
24. sec 1358
25. tan 2408
7p
28. cos }
4
8p
29. cot 2}
3
1
2
26. sin (21508)
27. csc (24208)
3p
30. tan 2}
4
11p
31. sec }
6
1
2
32. ERROR ANALYSIS Let (4, 3) be a point on the
terminal side of an angle u in standard position.
Describe and correct the error in finding tan u.
4
tan u 5 }x 5 }
y
3
33. ★ SHORT RESPONSE Write tan u as the ratio of two other trigonometric
functions. Use this ratio to explain why tan 908 is undefined but cot 908 5 0.
34. CHALLENGE Five of the most famous numbers in mathematics — 0, 1, π, e,
and i — are related by the simple equation eπi 1 1 5 0. Derive this equation
using Euler’s formula: ea 1 bi 5 ea (cos b 1 i sin b).
PROBLEM SOLVING
EXAMPLE 5
In Exercises 35 and 36, use the formula in Example 5 on page 869.
on p. 869
for Exs. 35–36
35. FOOTBALL You and a friend each kick a football with an initial speed of
49 feet per second. Your kick is projected at an angle of 458 and your friend’s
kick is projected at an angle of 608. About how much farther will your football
travel than your friend’s football?
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
36. IN-LINE SKATING At what speed must the in-line skater launch himself off
the ramp in order to land on the other side of the ramp?
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
EXAMPLE 6
on p. 870
for Exs. 37–38
37. ★ SHORT RESPONSE A Ferris wheel has a radius of 75 feet. You board a car at
the bottom of the Ferris wheel, which is 10 feet above the ground, and rotate
2558 counterclockwise before the ride temporarily stops. How high above
the ground are you when the ride stops? If the radius of the Ferris wheel is
doubled, is your height above the ground doubled? Explain.
13.3 Evaluate Trigonometric Functions of Any Angle
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38. MULTI-STEP PROBLEM When two atoms in a molecule
Y
are bonded to a common atom, chemists are interested
in both the bond angle and the lengths of the bonds. An
ozone molecule (O3) is made up of two oxygen atoms
bonded to a third oxygen atom, as shown.
XY
D
PM
a. In the diagram, coordinates are given in picometers
(pm). (Note: 1 pm 5 10212 m.) Find the coordinates
(x, y) of the center of the oxygen atom in Quadrant II.
X
PM
b. Find the distance d (in picometers) between the
centers of the two unbonded oxygen atoms.
39. ★ EXTENDED RESPONSE A sprinkler at ground level is used to water a garden.
The water leaving the sprinkler has an initial speed of 25 feet per second.
a. Calculate Copy the table below. Use the formula in Example 5 on
page 869 to complete the table.
Angle of sprinkler, u
258
308
358
408
458
508
558
608
658
Horizontal distance
water travels, d
?
?
?
?
?
?
?
?
?
b. Interpret What value of u appears to maximize the horizontal distance
traveled by the water? Use the formula for horizontal distance traveled
and the unit circle to explain why your answer makes sense.
c. Compare Compare the horizontal distance traveled by the water when
u 5 (45 2 k)8 with the distance when u 5 (45 1 k)8.
40. CHALLENGE The latitude of a point on Earth is the
degree measure of the shortest arc from that point to
the equator. For example, the latitude of point P in the
diagram equals the degree measure of arc PE. At what
latitude u is the circumference of the circle of latitude
at P half the distance around the equator?
Circle of
latitude
P
C
u
D
O
E
Equator
MIXED REVIEW
PREVIEW
Prepare for
Lesson 13.4
in Exs. 41–46.
Graph the function f. Then use the graph to determine whether the inverse of f
is a function. (p. 438)
41. f(x) 5 5x 1 2
42. f(x) 5 2x 1 7
43. f(x) 5 x 2 1 5
44. f(x) 5 4x2, x ≥ 0
45. f(x) 5 0.25x 2
46. f(x) 5 x 2 7
Find the range and standard deviation of the data set. (p. 744)
47. 3, 5, 2, 3, 7, 11, 8, 4
48. 18, 12, 15, 9, 13, 7, 4, 17
49. 5.9, 8.2, 3.7, 6.1, 2.9, 1.8, 5.7
50. 54, 60, 57, 53, 59, 51, 56, 62
Find the sum of the series.
15
51.
∑ (3i 1 2) (p. 802)
18
52.
i51
∑ 2(3)i 2 1 (p. 810)
i51
872
n2pe-1303.indd 872
24
53.
i51
5
54.
∑ (4i 1 1) (p. 802)
7
55.
∑ }14 1 }32 2
i51
PRACTICE
for Lesson
13.3, p. 1022
Chapter 13EXTRA
Trigonometric
Ratios and
Functions
∑ (17 2 2i) (p. 802)
i51
`
i21
(p. 810)
56.
∑ 8 1 }12 2
i21
(p. 820)
i51
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10/14/05 4:03:25 PM