Download The Tangent Function and Periodicity

Chapter 4
Lesson
Vocabulary
The Tangent Function
and Periodicity
4-6
tangent function
periodic function
period of a function
BIG IDEA
The sine and cosine functions are periodic,
repeating every 2π or 360º. The tangent function is periodic,
repeating every π or 180º.
A frieze pattern is a visual design
that repeats over and over along
a line. The frieze pattern at the
right appears on the Chan Chan
ruins in Trujillo, Peru.
Mental Math
How many times does the
minute hand of a clock
pass the number 6
between 10 A.M. and
6 P.M.?
In Lesson 4-5, you used values
of sine and cosine to graph
trigonometric functions. You
also observed that, like frieze
patterns, their graphs repeat as
you move horizontally. This
lesson extends those ideas to
the tangent function.
The Tangent Function
The correspondence θ → tan θ, when θ is a real number, defines the
sin θ
tangent function. From the definition tan θ = _____
, values for the
cos θ
tangent function can be generated.
Activity
Step 1 The table below contains some exact values of tan θ. It also shows
decimal equivalents of those values. Fill in the missing values.
252
θ
0
π
30º = __
π
45º = __
π
60º = __
tan θ
(exact)
0
√
3
___
3
1
√
3
undefined
?
?
?
0
tan θ
(approx.)
0
0.577
1
1.732
undefined
?
?
?
0
6
4
π 120º = ___
2π
3π
5π
90º = __
135º = ___
150º = ___
180º = π
4
3
6
3
2
θ
7π
210º = ___
6
5π
225º = ___
4
4π
240º = ___
3
3π
270º = ___
2
5π
300º = ___
3
7π
11π
315º = ___
330º = ____
360º = 2π
4
6
tan θ
(exact)
?
?
?
?
?
?
?
?
tan θ
(approx.)
?
?
?
?
?
?
?
?
Trigonometric Functions
SMP_SEFST_C04L06_252_256_FINAL.i252 252
4/30/09 3:56:22 PM
Lesson 4-6
Step 2 At the right, a graph of the values of the
tangent function given in the first part of
the table from Step 1 is shown. Copy this
graph, and add the points you found in
Step 1 to the graph.
Step 3 Draw a smooth curve through these points,
to show the graph of y = tan θ for all θ,
0º ≤ θ ≤ 360º, 0 ≤ θ ≤ 2π where
tan θ is defined.
y
2
1
θ
90˚
180˚
270˚
360˚
450˚
-1
-2
The Graph of the Tangent Function
3π
5π
At the right is a graph of y = tan x for – ___
≤ x ≤ ___
. Notice that this
2
2
graph looks strikingly different from the graphs of both the sine and
cosine functions. The tangent function has asymptotes and does not
have a maximum or minimum value.
y
h(x) = tan x
x
-270˚ -90˚
- 3π -π - π
2
2
90˚
π
2
270˚ 450˚
2π 5π
π 3π
2
2
Example 1
Consider f(x) = tan x.
a. Give the domain and range of the function f.
b. Is f an odd function, an even function, or neither? Justify your answer.
Solution
a. Because the tangent function has multiple vertical asymptotes, the
domain of the tangent function is the set of all real numbers
π
except odd multiples of 90º or __
. Notice that the tangent function
2
has no minimum or maximum values. Therefore, its range is the set of
all real numbers.
b. From the Opposites Theorem, tan(–x) = –tan x for all x.
Thus, the tangent function is an odd function.
Periodicity and the Trigonometric Functions
The periodic nature displayed by sine, cosine, and tangent is
summarized in the following theorem.
Periodicity Theorem
For any real number x and any integer n,
sin x = sin (x + n · 2π) = sin (x + n · 360º)
cos x = cos (x + n · 2π) = cos (x + n · 360º)
tan x = tan (x + n · π) = tan (x + n · 180º).
The Tangent Function and Periodicity
SMP_SEFST_C04L06_252_256_FINAL.i253 253
253
4/28/09 2:49:49 PM
SMP_SEFST_C04L06_252_256_FINAL.i254 Page 254 4/29/10 1:07:44 PM other
/Volumes/108/WG00060_r1/work%0/indd%0/SMP_FST_SE_......
Chapter 4
The theorem states that the sine and cosine functions are periodic
functions with period 360º or 2π radians, while the tangent function is a
periodic function with period 180o or π radians.
Definitions of Periodic and Period
A function f is periodic if there is a positive number p such that
f(x + p) = f(x) for all x in the domain of f. The smallest such p, if
it exists, is called the period of f.
A part of the function from any particular x to x + p, where p is the
period of the function, is called a cycle of the function. For instance, one
π
π
cycle of the tangent function is from 0 to π; another is from – __
to __
.
2
2
Example 2
Use the Periodicity Theorem to find cos 2670º.
2670º
Solution _____
360º ≈ 7.4, so 2670º – 7 · 360º will be less than 360º.
2670º – 7 · 360º = 150º, so R2670º = R150º. Therefore,
√
3
cos 2670º = cos 150º = – ___ .
2
Many phenomena are periodic, including tides, calendars, heart beats,
actions of circular gears, phases of the moon, and seasons of the year.
GUIDED
Example 3
a. the maximum and minimum values
b. the range
c. the period
Blood Pressure
y
Pressure
(mm mercury)
The graph at the right shows normal human blood
pressure as a function of time. Blood pressure is
systolic when the heart is contracting and diastolic
when the heart is expanding. The changes from
systolic to diastolic blood pressure create the pulse.
For this function, determine each.
Systolic
130
120
110
100
90
0
11.6
Solution
a. The maximum and minimum values of the graph are those values in which
the graph obtains a highest and lowest point, respectively. The maximum
value on this graph is ? , the minimum value is ? .
b. The range is the maximum value minus the minimum value. From Part a,
the range shown on this graph is ? .
c. The period is the range of x-values for the smallest section of the graph that
can be translated horizontally onto itself. The period shown on this
graph is ? seconds.
254
Trigonometric Functions
x
Diastolic
11.8
12
12.2
Time (seconds)
12.4
SMP_SEFST_C04L06_252_256_FINAL.i255 Page 255
1/9/10
4:18:59 PM /Volumes/121/WG00060/work%0/indd%0/SMP_FST_SE_C04/SMP_SEFST_C04L06_252_
u-s082
Lesson 4-6
Questions
COVERING THE IDEAS
1. a. List all values of θ between 0º and 360º such that cos θ = 0.
b. What is f(θ) = tan θ for these θ-values?
c. What do these values of θ mean for the graph of the
tangent function?
2. List all of the values of x from 0 to 2π for which sin x = 0. What do
these x-values indicate for the graph of the tangent function?
In 3–5, use the Periodicity Theorem to evaluate.
3. sin 495º
4. cos 810º
5. tan 3570º
4π
6. Given that tan ___
≈ 5.671, use the Periodicity Theorem to evaluate.
9
13π
5π
22π
a. tan ____
b. tan – ___
c. tan ____
9
9
9
7. What is the period of the function with the given equation?
a. y = sin x
b. y = cos x
c. y = tan x
APPLYING THE MATHEMATICS
8. Suppose that f is a periodic function whose domain is the real
6
numbers. One cycle of f is graphed at the right.
a. What is the period of f ?
y
4
f
2
b. Graph f on the interval –15 ≤ x ≤ 15.
c. Find f(51).
x
-5
d. Find four integer values of x such that f(x) = 0.
5
-2
9. If one endpoint of a cycle of the cosine function is 90º, where is the
other endpoint?
π
10. If one endpoint of a cycle of the tangent function is __
, where is the
2
other endpoint?
11. State equations for two of the asymptotes of the tangent function
a. in radians.
b. in degrees.
12. Let f(n) be the number in the nth decimal place of __17 .
a. Give the values of f(1), f(2), f(3), and f(4).
b. f is a periodic function. What is its period?
13. The table at the right contains hourly data for the height of tide
relative to the mean low water level in Pago Pago, American Samoa
on October 5, 2008.
a. Create a scatterplot of the data.
b. Determine the range of the data.
c. From the scatterplot, estimate the period of the data.
Hour
Height
Hour
Height
0
1.59
12
2.03
1
1.29
13
1.65
2
0.91
14
1.14
3
0.61
15
0.83
4
0.65
16
0.62
5
0.79
17
0.49
6
1.04
18
0.53
7
1.32
19
0.87
8
1.75
20
1.15
9
2.01
21
1.49
10
2.20
22
1.77
11
2.21
23
1.87
Source: National Oceanographic and
Atmospheric Administration
The Tangent Function and Periodicity
255
SMP_SEFST_C04L06_252_256_FINAL.i256 Page 256
1/9/10
4:19:07 PM /Volumes/121/WG00060/work%0/indd%0/SMP_FST_SE_C04/SMP_SEFST_C04L06_252_
u-s082
Chapter 4
REVIEW
14. Fill in the blanks for the graph of the sine function at
the right. (Lesson 4-5)
15. Refer to the predator-prey graph below. (Lesson 4-5)
( ? , 0)
(?, ?)
Population
For both the predator and the prey functions, determine the
a. domain.
b. maximum and minimum.
c. period.
1200
1100
1000
900
800
700
600
500
400
300
200
100
x
(3π, 0)
( ? , 0)
Prey
Predator
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48
Months
√
3
16. Find x such that 0 ≤ x ≤ 2π, if cos x = __12 and sin x = – ___
.
2
(Lesson 4-4)
17. Is the cosine function odd, even, or neither? Justify your
conclusion. (Lessons 4-5, 3-4)
18. Suppose Rθ(1, 0) = (-0.75, y) and is a point in Quadrant II. Find y.
(Lesson 4-3)
19. Under some translation T, the point (–6, 2) is mapped to (0, 7).
a. State the rule for T.
b. Find T(9, 9). (Lesson 3-2)
20. The graph at the right is from Weather on
T
40
Temperature (˚C)
the Planets, by George Ohring. It shows how
temperature is a function of latitude on Earth
and Mars when it is spring in one hemisphere
on each planet. Let L be the latitude on each
planet, E(L) be the average temperature at
latitude L on Earth, and M(L) be the average
temperature at latitude L on Mars.
a. Is L the dependent or independent variable?
b. Estimate E(60).
c. Estimate E(0) - M(0), and state what
quantity this expression represents.
d. What is the range of M? (Lesson 2-1)
20
Trigonometric Functions
Mars Average
T = M(L)
0
-40
-60
-80
L
-100
90
60
30
0
30
60
North Pole
21. The word tangent has another meaning in geometry. It also has
256
T = E(L)
-20
EXPLORATION
another meaning in English. What are these meanings?
Earth Average
90
South Pole
Latitude (Degrees)