Trigonometric Functions of Real Numbers; Periodic Functions
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526 Chapter 5 Trigonometric Functions
Writing in Mathematics 105. If you are given a point on the terminal side of angle u, explain how to find sin u. 106. Explain why tan 90° is undefined.
113. When I found the exact value of cos 14p 3 , I used a number of concepts, including coterminal angles, reference angles, finding the cosine of a special angle, and knowing the cosine’s sign in various quadrants.
107. If cos u 7 0 and tan u 6 0, explain how to find the quadrant in which u lies. 108. What is a reference angle? Give an example with your description. 109. Explain how reference angles are used to evaluate trigonometric functions. Give an example with your description.
Critical Thinking Exercises Make Sense? In Exercises 110–113, determine whether each statement makes sense or does not make sense, and explain your reasoning. 110. I’m working with a quadrantal angle u for which sin u is undefined. 111. This angle u is in a quadrant in which sin u 6 0 and csc u 7 0. 112. I am given that tan u = 35 , so I can conclude that y = 3 and x = 5.
Section
5.4
Objectives
Preview Exercises Exercises 114–116 will help you prepare for the material covered in the next section. 114. Graph: x2 + y2 = 1. Then locate the point A - 12 , graph.
B on the
115. Use your graph of x2 + y2 = 1 from Exercise 114 to determine the relation’s domain and range. 116. a. Find the exact value of sin
A p4 B , sin A - p4 B , sin A p3 B , and
sin A - B . Based on your results, can the sine function be an even function? Explain your answer. p 3
b. Find the exact value of cos cos A -
p 3
A p4 B , cos A - p4 B , cos A p3 B , and
B . Based on your results, can the cosine
function be an odd function? Explain your answer.
Trigonometric Functions of Real Numbers; Periodic Functions
Use a unit circle to define
13 2
C
ycles govern many aspects of life—heartbeats, sleep patterns, seasons, and tides all follow regular, predictable cycles. In this section, we will see why trigonometric functions are used to model phenomena that occur in cycles. To do this, we need to move beyond angles and consider trigonometric functions of real numbers.
trigonometric functions of real numbers. Recognize the domain and range of sine and cosine functions. Use even and odd trigonometric functions. Use periodic properties.
Use a unit circle to define trigonometric functions of real numbers.
Trigonometric Functions of Real Numbers
y
1
s t x (1, 0)
x2 + y2 = 1
Figure 5.53 Unit circle with a central angle measuring t radians
Thus far, we have considered trigonometric functions of angles measured in degrees or radians. To define trigonometric functions of real numbers, rather than angles, we use a unit circle. A unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this unit circle is x2 + y2 = 1. Figure 5.53 shows a unit circle in which the central angle measures t radians. We can use the formula for the length of a circular arc, s = ru, to find the length of the intercepted arc. s=ru=1 ⴢ t=t The radius of a unit circle is 1.
The radian measure of the central angle is t.
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Section 5.4 Trigonometric Functions of Real Numbers; Periodic Functions
527
Thus, the length of the intercepted arc is t. This is also the radian measure of the central angle. Thus, in a unit circle, the radian measure of the central angle is equal to the length of the intercepted arc. Both are given by the same real number t. In Figure 5.54, the radian measure of the angle and the length of the intercepted arc are both shown by t. Let P = 1x, y2 denote the point on the unit circle that has arc length t from (1, 0). Figure 5.54(a) shows that if t is positive, point P is reached by moving counterclockwise along the unit circle from (1, 0). Figure 5.54(b) shows that if t is negative, point P is reached by moving clockwise along the unit circle from (1, 0). For each real number t, there corresponds a point P = 1x, y2 on the unit circle. y
y P = (x, y)
x2 + y2 = 1
t t
(1, 0)
x
t
(1, 0)
t
x2 + y2 = 1
Figure 5.54
(a) t is positive.
(b) t is negative.
x = cos t and y = sin t.
t t x (1, 0)
(
P = (x, y)
Using Figure 5.54, we define the cosine function at t as the x-coordinate of P and the sine function at t as the y-coordinate of P. Thus,
y
P = − 35 , − 45
x
)
x2 + y2 = 1
For example, a point P = 1x, y2 on the unit circle corresponding to a real 3p . We see that the coordinates of number t is shown in Figure 5.55 for p 6 t 6 2 P = 1x, y2 are x = - 35 and y = - 45 . Because the cosine function is the x-coordinate of P and the sine function is the y-coordinate of P, the values of these trigonometric functions at the real number t are
Figure 5.55
cos t = -
3 5
and
4 sin t = - . 5
Definitions of the Trigonometric Functions in Terms of a Unit Circle
If t is a real number and P = 1x, y2 is a point on the unit circle that corresponds to t, then sin t = y
csc t =
1 ,y Z 0 y
cos t = x
sec t =
1 ,x Z 0 x
cot t =
x , y Z 0. y
tan t =
y ,x Z 0 x
Because this definition expresses function values in terms of coordinates of a point on a unit circle, the trigonometric functions are sometimes called the circular functions.
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528 Chapter 5 Trigonometric Functions EXAMPLE 1
In Figure 5.56, t is a real number equal to the length of the intercepted arc of an 1 23 angle that measures t radians and P = ¢ - , ≤ is a point on the unit circle that 2 2 corresponds to t. Use the figure to find the values of the trigonometric functions at t.
y
(
P = − 21 , 兹3 2
Finding Values of the Trigonometric Functions
) t t x
O
(1, 0)
Solution The point P on the unit circle that corresponds to t has coordinates 1 23 1 23 to find the values of the trigonometric ≤ . We use x = - and y = 2 2 2 2 functions.
¢- ,
x2 + y2 = 1
Figure 5.56
sin t = y =
23 2
csc t =
1 2
sec t =
cos t = x = -
23 y 2 = tan t = = - 23 x 1 2
Check Point
1 1 2 2 # 23 223 = = = = y 3 23 23 23 23 2 1 = x
1 = -2 1 2
1 x 2 1 1 # 23 23 = = cot t = = = y 3 23 23 23 23 2
1
y
Use the figure on the right to find the values of the trigonometric functions at t.
(
P = 兹3 , 21 2
t O
)
t x (1, 0)
x2 + y2 = 1
EXAMPLE 2 y
Finding Values of the Trigonometric Functions
Use Figure 5.57 to find the values of the trigonometric functions at t = P = (0, 1) p 2
p has coordinates 2 p (0, 1). We use x = 0 and y = 1 to find the values of the trigonometric functions at . 2
Solution The point P on the unit circle that corresponds to t =
p 2
x (1, 0)
x2 + y2 = 1
p . 2
sin
p =y=1 2
cos
p =x=0 2
tan
y 1 p = x= 0 2
Figure 5.57
sec p and 2 tan p are 2 undefined.
csc
p 1 1 = = =1 2 y 1
sec
p 1 1 = = 2 0 x
cot
x p 0 = y = =0 2 1
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Section 5.4 Trigonometric Functions of Real Numbers; Periodic Functions
Check Point
y
2
Use the figure on the right to find the values of the trigonometric functions at t = p.
529
p P = (−1, 0) p x
x2 + y2 = 1
Recognize the domain and range of sine and cosine functions.
Domain and Range of Sine and Cosine Functions The value of a trigonometric function at the real number t is its value at an angle of t radians. However, using real number domains, we can observe properties of trigonometric functions that are not as apparent using the angle approach. For example, the domain and range of each trigonometric function can be found from the unit circle definition. At this point, let’s look only at the sine and cosine functions, sin t = y and
Figure 5.58 shows the sine function at t as the y-coordinate of a point along the unit circle:
y y = sin t
(0, 1)
y=sin t.
(x, y) 1 t
t x
The domain is associated with t, the angle’s radian measure and the intercepted arc’s length. The range is associated with y, the point's second coordinate.
(0, 1)
Because t can be any real number, the domain of the sine function is 1 - q , q 2, the set of all real numbers. The radius of the unit circle is 1 and the dashed horizontal lines in Figure 5.58 show that y cannot be less than -1 or greater than 1. Thus, the range of the sine function is 3-1, 14, the set of all real numbers from - 1 to 1, inclusive. Figure 5.59 shows the cosine function at t as the x-coordinate of a point along the unit circle:
Figure 5.58
y x = cos t (x, y) 1
x=cos t.
t t
(1, 0)
cos t = x.
(1, 0)
x
The domain is associated with t, the angle’s radian measure and the intercepted arc’s length. The range is associated with x, the point's first coordinate.
Figure 5.59
Because t can be any real number, the domain of the cosine function is 1- q , q 2. The radius of the unit circle is 1 and the dashed vertical lines in Figure 5.59 show that x cannot be less than -1 or greater than 1. Thus, the range of the cosine function is 3- 1, 14.
The Domain and Range of the Sine and Cosine Functions
The domain of the sine function and the cosine function is 1 - q , q 2, the set of all real numbers. The range of these functions is 3-1, 14, the set of all real numbers from -1 to 1, inclusive.
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530 Chapter 5 Trigonometric Functions
Even and Odd Trigonometric Functions
Use even and odd trigonometric functions.
We have seen that a function is even if f1- t2 = f1t2 and odd if f1 -t2 = - f1t2. We can use Figure 5.60 to show that the cosine function is an even function and the sine function is an odd function. By definition, the coordinates of the points P and Q in Figure 5.60 are as follows: P: 1cos t, sin t2 Q: 1cos1- t2, sin1 -t22.
y x2 + y2 = 1
In Figure 5.60, the x-coordinates of P and Q are the same. Thus,
P t
cos1- t2 = cot t. x
O −t
This shows that the cosine function is an even function. By contrast, the y-coordinates of P and Q are negatives of each other. Thus,
Q
sin1-t2 = - sin t. Figure 5.60
This shows that the sine function is an odd function. This argument is valid regardless of the length of t. Thus, the arc may terminate in any of the four quadrants or on any axis. Using the unit circle definition of the trigonometric functions, we obtain the following results:
Even and Odd Trigonometric Functions The cosine and secant functions are even. cos1-t2 = cos t
sec1-t2 = sec t
The sine, cosecant, tangent, and cotangent functions are odd. sin1- t2 = - sin t
csc1 - t2 = - csc t
tan1 -t2 = - tan t
cot1-t2 = - cot t
Using Even and Odd Functions to Find Exact Values
EXAMPLE 3
Find the exact value of each trigonometric function: p a. cos1 - 45°2 b. tan a - b. 3
Solution a. cos1 -45°2 = cos 45° = b. tan a -
22 2
p p b = - tan = - 23 3 3
Check Point
3
a. cos1 -60°2
Find the exact value of each trigonometric function: b. tan a -
p b. 6
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Section 5.4 Trigonometric Functions of Real Numbers; Periodic Functions
Use periodic properties.
531
Periodic Functions Certain patterns in nature repeat again and again. For example, the ocean level at a beach varies from low tide to high tide and then back to low tide approximately every 12 hours. If low tide occurs at noon, then high tide will be around 6 P.M. and low tide will occur again around midnight, and so on infinitely. If f1t2 represents the ocean level at the beach at any time t, then the level is the same 12 hours later. Thus, f1t + 122 = f1t2. The word periodic means that this tidal behavior repeats infinitely. The period, 12 hours, is the time it takes to complete one full cycle.
Definition of a Periodic Function A function f is periodic if there exists a positive number p such that f1t + p2 = f1t2 for all t in the domain of f. The smallest positive number p for which f is periodic is called the period of f.
The trigonometric functions are used to model periodic phenomena.Why? If we begin at any point P on the unit circle and travel a distance of 2p units along the perimeter, we will return to the same point P. Because the trigonometric functions are defined in terms of the coordinates of that point P, we obtain the following results:
Periodic Properties of the Sine and Cosine Functions sin1t + 2p2 = sin t and
cos1t + 2p2 = cos t
The sine and cosine functions are periodic functions and have period 2p.
Using Periodic Properties to Find Exact Values
EXAMPLE 4
Find the exact value of each trigonometric function: 9p a. cos 420° b. sin . 4
Solution a. cos 420° = cos 160° + 360°2 = cos 60° = b. sin
1 2
9p p p 22 = sin a + 2pb = sin = 4 4 4 2
Check Point a. cos 405°
4
Find the exact value of each trigonometric function: b. sin
7p . 3
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532 Chapter 5 Trigonometric Functions Like the sine and cosine functions, the secant and cosecant functions have period 2p. However, the tangent and cotangent functions have a smaller period. Figure 5.61 shows that if we begin at any point P1x, y2 on the unit circle and travel a distance of p units along the perimeter, we arrive at the point Q1 -x, -y2. The tangent function, defined in terms of the coordinates of a point, is the same at 1x, y2 and 1-x, -y2.
p
y its un
P(x, y) t
t p t
x
Tangent function at (x, y)
y –y = x –x
Tangent function p radians later
Q(x, y)
We see that tan1t + p2 = tan t. The same observations apply to the cotangent function.
Figure 5.61 tangent at P = tangent at Q
Periodic Properties of the Tangent and Cotangent Functions tan1t + p2 = tan t and cot1t + p2 = cot t The tangent and cotangent functions are periodic functions and have period p.
Why do the trigonometric functions model phenomena that repeat indefinitely? By starting at point P on the unit circle and traveling a distance of 2p units, 4p units, 6p units, and so on, we return to the starting point P. Because the trigonometric functions are defined in terms of the coordinates of that point P, if we add (or subtract) multiples of 2p to t, the values of the trigonometric functions of t do not change. Furthermore, the values for the tangent and cotangent functions of t do not change if we add (or subtract) multiples of p to t.
Repetitive Behavior of the Sine, Cosine, and Tangent Functions For any integer n and real number t, sin1t + 2pn2 = sin t, cos1t + 2pn2 = cos t, and tan1t + pn2 = tan t.
Exercise Set 5.4 Practice Exercises In Exercises 1–4, a point P1x, y2 is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t. 1.
2.
y
(
15 8 P − 17 , 17
)
y
t
t
t O
t x
O
(1, 0)
(
5 12 P − 13 , − 13
)
x (1, 0)
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Section 5.4 Trigonometric Functions of Real Numbers; Periodic Functions 3.
In Exercises 19–24, a. Use the unit circle shown for Exercises 5–18 to find the value of the trigonometric function.
y
O
b. Use even and odd properties of trigonometric functions and your answer from part (a) to find the value of the same trigonometric function at the indicated real number. p p 20. a. cos 19. a. cos 3 6 p p b. cos a - b b. cos a - b 3 6
x
t
(1, 0) t
(
兹2 兹2 P 2 ,− 2
)
21. a. sin 4. P
(
,
兹2 2
5p 6
22. a. sin 5p b 6
b. sin a -
y − 兹2 2
533
) t
23. a. tan
t x
O
b. sin a -
2p b 3
11p 6 11p b. tan a b 6
5p 3
b. tan a -
2p 3
24. a. tan 5p b 3
(1, 0)
In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of 5p 3p 7p p p 3p , p, , , , and 2p. 0, , , 4 2 4 4 2 4
In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of 0,
p p p 2p 5p 7p 4p 3p 5p 11p , , , , , p, , , , , , and 2p. 6 3 2 3 6 6 3 2 3 6
a. Use the 1x, y2 coordinates in the figure to find the value of the trigonometric function. b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number. y
Use the 1x, y2 coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.
(−
兹2 2
, 兹2 2
)
(0, 1) f
y
( (
− 兹3 2
,
1 兹3 , 2 2
−
1 2
)
(
兹2 2
(
(0, 1)
1 兹3 , 2 2
u
)
(
k
)
兹3 2
x
(−1, 0)
,
1 2
(−
)
兹2 2
, − 兹2 2
)
(1, 0)
(
(0, −1)
兹2 2
x 兹3 2
, − 21
(− 5. sin
p 6
2p 8. cos 3
1 , 2
(1, 0)
)
−
(0, −1) 兹3 2
)
( 6. sin
p 3
9. tan p
)
d
(−1, 0)
(−
, 兹2 2
1 , 2
− 兹3 2
( )
兹3 2
, − 21
)
7. cos
3p 4 11p b. sin 4 p 27. a. cos 2 9p b. cos 2
25. a. sin
5p 6
10. tan 0
11. csc
7p 6
12. csc
4p 3
13. sec
11p 6
14. sec
5p 3
15. sin
3p 2
16. cos
3p 2
17. sec
3p 2
18. tan
3p 2
29. a. tan p b. tan 17p
31. a.
sin
b. sin
7p 4 47p 4
3p 4 11p cos 4 p sin 2 9p sin 2 p cot 2 15p cot 2
26. a. cos b. 28. a. b. 30. a. b. 32. a.
cos
b. cos
7p 4 47p 4
, − 兹2 2
)
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534 Chapter 5 Trigonometric Functions
Practice Plus In Exercises 33–42, let sin t = a, cos t = b, and tan t = c. Write each expression in terms of a, b, and c. 33. sin1 -t2 - sin t
34. tan1 -t2 - tan t
35. 4 cos1 -t2 - cos t
36. 3 cos1 -t2 - cos t
1 representing peak emotional well-being, - 1 representing the low for emotional well-being, and 0 representing feeling neither emotionally high nor low. a. Find E corresponding to t = 7, 14, 21, 28, and 35. Describe what you observe. b. What is the period of the emotional cycle?
41. cos t + cos1t + 1000p2 - tan t - tan1t + 999p2 - sin t + 4 sin1t - 1000p2
46. The height of the water, H, in feet, at a boat dock t hours after 6 A.M. is given by p H = 10 + 4 sin t. 6 a. Find the height of the water at the dock at 6 A.M., 9 A.M., noon, 6 P.M., midnight, and 3 A.M. b. When is low tide and when is high tide? c. What is the period of this function and what does this mean about the tides?
42. - cos t + 7 cos1t + 1000p2 + tan t + tan1t + 999p2 + sin t + sin1t - 1000p2
Writing in Mathematics
37. sin1t + 2p2 - cos1t + 4p2 + tan1t + p2 38. sin1t + 2p2 + cos1t + 4p2 - tan1t + p2 39. sin1 -t - 2p2 - cos1 -t - 4p2 - tan1 -t - p2 40. sin1 - t - 2p2 + cos1 - t - 4p2 - tan1 - t - p2
Application Exercises 43. The number of hours of daylight, H, on day t of any given year (on January 1, t = 1) in Fairbanks, Alaska, can be modeled by the function H1t2 = 12 + 8.3 sin c
2p 1t - 802 d . 365
a. March 21, the 80th day of the year, is the spring equinox. Find the number of hours of daylight in Fairbanks on this day. b. June 21, the 172nd day of the year, is the summer solstice, the day with the maximum number of hours of daylight. To the nearest tenth of an hour, find the number of hours of daylight in Fairbanks on this day.
47. Why are the trigonometric functions sometimes called circular functions? 48. What is the range of the sine function? Use the unit circle to explain where this range comes from. 49. What do we mean by even trigonometric functions? Which of the six functions fall into this category? 50. What is a periodic function? Why are the sine and cosine functions periodic? 51. Explain how you can use the function for emotional fluctuations in Exercise 45 to determine good days for having dinner with your moody boss. 52. Describe a phenomenon that repeats infinitely. What is its period?
Critical Thinking Exercises
c. December 21, the 355th day of the year, is the winter solstice, the day with the minimum number of hours of daylight. Find, to the nearest tenth of an hour, the number of hours of daylight in Fairbanks on this day.
Make Sense? In Exercises 53–56, determine whether each statement makes sense or does not make sense, and explain your reasoning.
44. The number of hours of daylight, H, on day t of any given year (on January 1, t = 1) in San Diego, California, can be modeled by the function
53. Assuming that the innermost circle on this Navajo sand painting is a unit circle, as A moves around the circle, its coordinates define the cosine and sine functions, respectively.
H1t2 = 12 + 2.4 sin c
2p 1t - 802 d . 365
a. March 21, the 80th day of the year, is the spring equinox. Find the number of hours of daylight in San Diego on this day.
A
b. June 21, the 172nd day of the year, is the summer solstice, the day with the maximum number of hours of daylight. Find, to the nearest tenth of an hour, the number of hours of daylight in San Diego on this day. c. December 21, the 355th day of the year, is the winter solstice, the day with the minimum number of hours of daylight. To the nearest tenth of an hour, find the number of hours of daylight in San Diego on this day. 45. People who believe in biorhythms claim that there are three cycles that rule our behavior—the physical, emotional, and mental. Each is a sine function of a certain period. The function for our emotional fluctuations is p E = sin t, 14 where t is measured in days starting at birth. Emotional fluctuations, E, are measured from - 1 to 1, inclusive, with
54. I’m using a value for t and a point on the unit circle corresponding to t for which sin t = -
110 . 2
p 23 p 13 = , I can conclude that cos a- b = . 6 2 6 2 7p 56. I can find the exact value of sin 3 using periodic properties of the sine function, or using a coterminal angle and a reference angle. 55. Because cos
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Mid-Chapter Check Point 57. Find the exact value of cos 0° + cos 1° + cos 2° + cos 3° + Á + cos 179° + cos 180°.
62. y = 4 sin A 2x x
58. If f1x2 = sin x and f1a2 = 14 , find the value of f1a2 + f1a + 2p2 + f1a + 4p2 + f1a + 6p2. 59. If f1x2 = sin x f1a2 + 2f1 - a2.
and
f1a2 = 14 ,
find
the
value
p 3
2p 3
535
B
7p 12
13p 12
5p 6
4p 3
y of
60. The seats of a Ferris wheel are 40 feet from the wheel’s center. When you get on the ride, your seat is 5 feet above the ground. How far above the ground are you after rotating through an angle of 765°? Round to the nearest foot.
63. y = 3 sin p2 x x
0
1 3
5 3
1
2
7 3
11 3
3
4
y
Preview Exercises Exercises 61–63 will help you prepare for the material covered in the next section. In each exercise, complete the table of coordinates. Do not use a calculator.
After completing this table of coordinates, plot the nine ordered pairs as points in a rectangular coordinate system. Then connect the points with a smooth curve.
61. y = 12 cos 14x + p2 x
- p4
- p8
0
p 8
p 4
y
Chapter
5
Mid-Chapter Check Point
What You Know: We learned to use radians to measure angles: One radian (approximately 57°) is the measure of the central angle that intercepts an arc equal in length to the radius of the circle. Using 180° = p radians, p b and we converted degrees to radians a multiply by 180° 180° b . We defined the six radians to degrees a multiply by p trigonometric functions using right triangles, angles in standard position, and coordinates of points along the unit circle. Evaluating trigonometric functions using reference angles involved connecting a number of concepts, including finding coterminal and reference angles, locating special angles, determining the signs of the trigonometric functions in specific quadrants, and finding the function values at special angles. Use the important Study Tip on page 524 as a reference sheet to help connect these concepts.
In Exercises 5–7, a. Find a positive angle less than 360° or 2p that is coterminal with the given angle. b. Draw the given angle in standard position. c. Find the reference angle for the given angle. 5.
11p 3
6. -
7. 510°
8. Use the triangle to find each of the six trigonometric functions of u. B
6
5
u A
C
9. Use the point on the terminal side of u to find each of the six trigonometric functions of u.
In Exercises 1–2, convert each angle in degrees to radians. Express your answer as a multiple of p. 1. 10°
19p 4
y
2. -105° u x
In Exercises 3–4, convert each angle in radians to degrees. 3.
5p 12
4. -
13p 20
P(3, ⫺2)
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526 Chapter 5 Trigonometric Functions
Writing in Mathematics 105. If you are given a point on the terminal side of angle u, explain how to find sin u. 106. Explain why tan 90° is undefined.
113. When I found the exact value of cos 14p 3 , I used a number of concepts, including coterminal angles, reference angles, finding the cosine of a special angle, and knowing the cosine’s sign in various quadrants.
107. If cos u 7 0 and tan u 6 0, explain how to find the quadrant in which u lies. 108. What is a reference angle? Give an example with your description. 109. Explain how reference angles are used to evaluate trigonometric functions. Give an example with your description.
Critical Thinking Exercises Make Sense? In Exercises 110–113, determine whether each statement makes sense or does not make sense, and explain your reasoning. 110. I’m working with a quadrantal angle u for which sin u is undefined. 111. This angle u is in a quadrant in which sin u 6 0 and csc u 7 0. 112. I am given that tan u = 35 , so I can conclude that y = 3 and x = 5.
Section
5.4
Objectives
Preview Exercises Exercises 114–116 will help you prepare for the material covered in the next section. 114. Graph: x2 + y2 = 1. Then locate the point A - 12 , graph.
B on the
115. Use your graph of x2 + y2 = 1 from Exercise 114 to determine the relation’s domain and range. 116. a. Find the exact value of sin
A p4 B , sin A - p4 B , sin A p3 B , and
sin A - B . Based on your results, can the sine function be an even function? Explain your answer. p 3
b. Find the exact value of cos cos A -
p 3
A p4 B , cos A - p4 B , cos A p3 B , and
B . Based on your results, can the cosine
function be an odd function? Explain your answer.
Trigonometric Functions of Real Numbers; Periodic Functions
Use a unit circle to define
13 2
C
ycles govern many aspects of life—heartbeats, sleep patterns, seasons, and tides all follow regular, predictable cycles. In this section, we will see why trigonometric functions are used to model phenomena that occur in cycles. To do this, we need to move beyond angles and consider trigonometric functions of real numbers.
trigonometric functions of real numbers. Recognize the domain and range of sine and cosine functions. Use even and odd trigonometric functions. Use periodic properties.
Use a unit circle to define trigonometric functions of real numbers.
Trigonometric Functions of Real Numbers
y
1
s t x (1, 0)
x2 + y2 = 1
Figure 5.53 Unit circle with a central angle measuring t radians
Thus far, we have considered trigonometric functions of angles measured in degrees or radians. To define trigonometric functions of real numbers, rather than angles, we use a unit circle. A unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this unit circle is x2 + y2 = 1. Figure 5.53 shows a unit circle in which the central angle measures t radians. We can use the formula for the length of a circular arc, s = ru, to find the length of the intercepted arc. s=ru=1 ⴢ t=t The radius of a unit circle is 1.
The radian measure of the central angle is t.
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Section 5.4 Trigonometric Functions of Real Numbers; Periodic Functions
527
Thus, the length of the intercepted arc is t. This is also the radian measure of the central angle. Thus, in a unit circle, the radian measure of the central angle is equal to the length of the intercepted arc. Both are given by the same real number t. In Figure 5.54, the radian measure of the angle and the length of the intercepted arc are both shown by t. Let P = 1x, y2 denote the point on the unit circle that has arc length t from (1, 0). Figure 5.54(a) shows that if t is positive, point P is reached by moving counterclockwise along the unit circle from (1, 0). Figure 5.54(b) shows that if t is negative, point P is reached by moving clockwise along the unit circle from (1, 0). For each real number t, there corresponds a point P = 1x, y2 on the unit circle. y
y P = (x, y)
x2 + y2 = 1
t t
(1, 0)
x
t
(1, 0)
t
x2 + y2 = 1
Figure 5.54
(a) t is positive.
(b) t is negative.
x = cos t and y = sin t.
t t x (1, 0)
(
P = (x, y)
Using Figure 5.54, we define the cosine function at t as the x-coordinate of P and the sine function at t as the y-coordinate of P. Thus,
y
P = − 35 , − 45
x
)
x2 + y2 = 1
For example, a point P = 1x, y2 on the unit circle corresponding to a real 3p . We see that the coordinates of number t is shown in Figure 5.55 for p 6 t 6 2 P = 1x, y2 are x = - 35 and y = - 45 . Because the cosine function is the x-coordinate of P and the sine function is the y-coordinate of P, the values of these trigonometric functions at the real number t are
Figure 5.55
cos t = -
3 5
and
4 sin t = - . 5
Definitions of the Trigonometric Functions in Terms of a Unit Circle
If t is a real number and P = 1x, y2 is a point on the unit circle that corresponds to t, then sin t = y
csc t =
1 ,y Z 0 y
cos t = x
sec t =
1 ,x Z 0 x
cot t =
x , y Z 0. y
tan t =
y ,x Z 0 x
Because this definition expresses function values in terms of coordinates of a point on a unit circle, the trigonometric functions are sometimes called the circular functions.
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528 Chapter 5 Trigonometric Functions EXAMPLE 1
In Figure 5.56, t is a real number equal to the length of the intercepted arc of an 1 23 angle that measures t radians and P = ¢ - , ≤ is a point on the unit circle that 2 2 corresponds to t. Use the figure to find the values of the trigonometric functions at t.
y
(
P = − 21 , 兹3 2
Finding Values of the Trigonometric Functions
) t t x
O
(1, 0)
Solution The point P on the unit circle that corresponds to t has coordinates 1 23 1 23 to find the values of the trigonometric ≤ . We use x = - and y = 2 2 2 2 functions.
¢- ,
x2 + y2 = 1
Figure 5.56
sin t = y =
23 2
csc t =
1 2
sec t =
cos t = x = -
23 y 2 = tan t = = - 23 x 1 2
Check Point
1 1 2 2 # 23 223 = = = = y 3 23 23 23 23 2 1 = x
1 = -2 1 2
1 x 2 1 1 # 23 23 = = cot t = = = y 3 23 23 23 23 2
1
y
Use the figure on the right to find the values of the trigonometric functions at t.
(
P = 兹3 , 21 2
t O
)
t x (1, 0)
x2 + y2 = 1
EXAMPLE 2 y
Finding Values of the Trigonometric Functions
Use Figure 5.57 to find the values of the trigonometric functions at t = P = (0, 1) p 2
p has coordinates 2 p (0, 1). We use x = 0 and y = 1 to find the values of the trigonometric functions at . 2
Solution The point P on the unit circle that corresponds to t =
p 2
x (1, 0)
x2 + y2 = 1
p . 2
sin
p =y=1 2
cos
p =x=0 2
tan
y 1 p = x= 0 2
Figure 5.57
sec p and 2 tan p are 2 undefined.
csc
p 1 1 = = =1 2 y 1
sec
p 1 1 = = 2 0 x
cot
x p 0 = y = =0 2 1
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Section 5.4 Trigonometric Functions of Real Numbers; Periodic Functions
Check Point
y
2
Use the figure on the right to find the values of the trigonometric functions at t = p.
529
p P = (−1, 0) p x
x2 + y2 = 1
Recognize the domain and range of sine and cosine functions.
Domain and Range of Sine and Cosine Functions The value of a trigonometric function at the real number t is its value at an angle of t radians. However, using real number domains, we can observe properties of trigonometric functions that are not as apparent using the angle approach. For example, the domain and range of each trigonometric function can be found from the unit circle definition. At this point, let’s look only at the sine and cosine functions, sin t = y and
Figure 5.58 shows the sine function at t as the y-coordinate of a point along the unit circle:
y y = sin t
(0, 1)
y=sin t.
(x, y) 1 t
t x
The domain is associated with t, the angle’s radian measure and the intercepted arc’s length. The range is associated with y, the point's second coordinate.
(0, 1)
Because t can be any real number, the domain of the sine function is 1 - q , q 2, the set of all real numbers. The radius of the unit circle is 1 and the dashed horizontal lines in Figure 5.58 show that y cannot be less than -1 or greater than 1. Thus, the range of the sine function is 3-1, 14, the set of all real numbers from - 1 to 1, inclusive. Figure 5.59 shows the cosine function at t as the x-coordinate of a point along the unit circle:
Figure 5.58
y x = cos t (x, y) 1
x=cos t.
t t
(1, 0)
cos t = x.
(1, 0)
x
The domain is associated with t, the angle’s radian measure and the intercepted arc’s length. The range is associated with x, the point's first coordinate.
Figure 5.59
Because t can be any real number, the domain of the cosine function is 1- q , q 2. The radius of the unit circle is 1 and the dashed vertical lines in Figure 5.59 show that x cannot be less than -1 or greater than 1. Thus, the range of the cosine function is 3- 1, 14.
The Domain and Range of the Sine and Cosine Functions
The domain of the sine function and the cosine function is 1 - q , q 2, the set of all real numbers. The range of these functions is 3-1, 14, the set of all real numbers from -1 to 1, inclusive.
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530 Chapter 5 Trigonometric Functions
Even and Odd Trigonometric Functions
Use even and odd trigonometric functions.
We have seen that a function is even if f1- t2 = f1t2 and odd if f1 -t2 = - f1t2. We can use Figure 5.60 to show that the cosine function is an even function and the sine function is an odd function. By definition, the coordinates of the points P and Q in Figure 5.60 are as follows: P: 1cos t, sin t2 Q: 1cos1- t2, sin1 -t22.
y x2 + y2 = 1
In Figure 5.60, the x-coordinates of P and Q are the same. Thus,
P t
cos1- t2 = cot t. x
O −t
This shows that the cosine function is an even function. By contrast, the y-coordinates of P and Q are negatives of each other. Thus,
Q
sin1-t2 = - sin t. Figure 5.60
This shows that the sine function is an odd function. This argument is valid regardless of the length of t. Thus, the arc may terminate in any of the four quadrants or on any axis. Using the unit circle definition of the trigonometric functions, we obtain the following results:
Even and Odd Trigonometric Functions The cosine and secant functions are even. cos1-t2 = cos t
sec1-t2 = sec t
The sine, cosecant, tangent, and cotangent functions are odd. sin1- t2 = - sin t
csc1 - t2 = - csc t
tan1 -t2 = - tan t
cot1-t2 = - cot t
Using Even and Odd Functions to Find Exact Values
EXAMPLE 3
Find the exact value of each trigonometric function: p a. cos1 - 45°2 b. tan a - b. 3
Solution a. cos1 -45°2 = cos 45° = b. tan a -
22 2
p p b = - tan = - 23 3 3
Check Point
3
a. cos1 -60°2
Find the exact value of each trigonometric function: b. tan a -
p b. 6
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Section 5.4 Trigonometric Functions of Real Numbers; Periodic Functions
Use periodic properties.
531
Periodic Functions Certain patterns in nature repeat again and again. For example, the ocean level at a beach varies from low tide to high tide and then back to low tide approximately every 12 hours. If low tide occurs at noon, then high tide will be around 6 P.M. and low tide will occur again around midnight, and so on infinitely. If f1t2 represents the ocean level at the beach at any time t, then the level is the same 12 hours later. Thus, f1t + 122 = f1t2. The word periodic means that this tidal behavior repeats infinitely. The period, 12 hours, is the time it takes to complete one full cycle.
Definition of a Periodic Function A function f is periodic if there exists a positive number p such that f1t + p2 = f1t2 for all t in the domain of f. The smallest positive number p for which f is periodic is called the period of f.
The trigonometric functions are used to model periodic phenomena.Why? If we begin at any point P on the unit circle and travel a distance of 2p units along the perimeter, we will return to the same point P. Because the trigonometric functions are defined in terms of the coordinates of that point P, we obtain the following results:
Periodic Properties of the Sine and Cosine Functions sin1t + 2p2 = sin t and
cos1t + 2p2 = cos t
The sine and cosine functions are periodic functions and have period 2p.
Using Periodic Properties to Find Exact Values
EXAMPLE 4
Find the exact value of each trigonometric function: 9p a. cos 420° b. sin . 4
Solution a. cos 420° = cos 160° + 360°2 = cos 60° = b. sin
1 2
9p p p 22 = sin a + 2pb = sin = 4 4 4 2
Check Point a. cos 405°
4
Find the exact value of each trigonometric function: b. sin
7p . 3
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532 Chapter 5 Trigonometric Functions Like the sine and cosine functions, the secant and cosecant functions have period 2p. However, the tangent and cotangent functions have a smaller period. Figure 5.61 shows that if we begin at any point P1x, y2 on the unit circle and travel a distance of p units along the perimeter, we arrive at the point Q1 -x, -y2. The tangent function, defined in terms of the coordinates of a point, is the same at 1x, y2 and 1-x, -y2.
p
y its un
P(x, y) t
t p t
x
Tangent function at (x, y)
y –y = x –x
Tangent function p radians later
Q(x, y)
We see that tan1t + p2 = tan t. The same observations apply to the cotangent function.
Figure 5.61 tangent at P = tangent at Q
Periodic Properties of the Tangent and Cotangent Functions tan1t + p2 = tan t and cot1t + p2 = cot t The tangent and cotangent functions are periodic functions and have period p.
Why do the trigonometric functions model phenomena that repeat indefinitely? By starting at point P on the unit circle and traveling a distance of 2p units, 4p units, 6p units, and so on, we return to the starting point P. Because the trigonometric functions are defined in terms of the coordinates of that point P, if we add (or subtract) multiples of 2p to t, the values of the trigonometric functions of t do not change. Furthermore, the values for the tangent and cotangent functions of t do not change if we add (or subtract) multiples of p to t.
Repetitive Behavior of the Sine, Cosine, and Tangent Functions For any integer n and real number t, sin1t + 2pn2 = sin t, cos1t + 2pn2 = cos t, and tan1t + pn2 = tan t.
Exercise Set 5.4 Practice Exercises In Exercises 1–4, a point P1x, y2 is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t. 1.
2.
y
(
15 8 P − 17 , 17
)
y
t
t
t O
t x
O
(1, 0)
(
5 12 P − 13 , − 13
)
x (1, 0)
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Section 5.4 Trigonometric Functions of Real Numbers; Periodic Functions 3.
In Exercises 19–24, a. Use the unit circle shown for Exercises 5–18 to find the value of the trigonometric function.
y
O
b. Use even and odd properties of trigonometric functions and your answer from part (a) to find the value of the same trigonometric function at the indicated real number. p p 20. a. cos 19. a. cos 3 6 p p b. cos a - b b. cos a - b 3 6
x
t
(1, 0) t
(
兹2 兹2 P 2 ,− 2
)
21. a. sin 4. P
(
,
兹2 2
5p 6
22. a. sin 5p b 6
b. sin a -
y − 兹2 2
533
) t
23. a. tan
t x
O
b. sin a -
2p b 3
11p 6 11p b. tan a b 6
5p 3
b. tan a -
2p 3
24. a. tan 5p b 3
(1, 0)
In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of 5p 3p 7p p p 3p , p, , , , and 2p. 0, , , 4 2 4 4 2 4
In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of 0,
p p p 2p 5p 7p 4p 3p 5p 11p , , , , , p, , , , , , and 2p. 6 3 2 3 6 6 3 2 3 6
a. Use the 1x, y2 coordinates in the figure to find the value of the trigonometric function. b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number. y
Use the 1x, y2 coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.
(−
兹2 2
, 兹2 2
)
(0, 1) f
y
( (
− 兹3 2
,
1 兹3 , 2 2
−
1 2
)
(
兹2 2
(
(0, 1)
1 兹3 , 2 2
u
)
(
k
)
兹3 2
x
(−1, 0)
,
1 2
(−
)
兹2 2
, − 兹2 2
)
(1, 0)
(
(0, −1)
兹2 2
x 兹3 2
, − 21
(− 5. sin
p 6
2p 8. cos 3
1 , 2
(1, 0)
)
−
(0, −1) 兹3 2
)
( 6. sin
p 3
9. tan p
)
d
(−1, 0)
(−
, 兹2 2
1 , 2
− 兹3 2
( )
兹3 2
, − 21
)
7. cos
3p 4 11p b. sin 4 p 27. a. cos 2 9p b. cos 2
25. a. sin
5p 6
10. tan 0
11. csc
7p 6
12. csc
4p 3
13. sec
11p 6
14. sec
5p 3
15. sin
3p 2
16. cos
3p 2
17. sec
3p 2
18. tan
3p 2
29. a. tan p b. tan 17p
31. a.
sin
b. sin
7p 4 47p 4
3p 4 11p cos 4 p sin 2 9p sin 2 p cot 2 15p cot 2
26. a. cos b. 28. a. b. 30. a. b. 32. a.
cos
b. cos
7p 4 47p 4
, − 兹2 2
)
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534 Chapter 5 Trigonometric Functions
Practice Plus In Exercises 33–42, let sin t = a, cos t = b, and tan t = c. Write each expression in terms of a, b, and c. 33. sin1 -t2 - sin t
34. tan1 -t2 - tan t
35. 4 cos1 -t2 - cos t
36. 3 cos1 -t2 - cos t
1 representing peak emotional well-being, - 1 representing the low for emotional well-being, and 0 representing feeling neither emotionally high nor low. a. Find E corresponding to t = 7, 14, 21, 28, and 35. Describe what you observe. b. What is the period of the emotional cycle?
41. cos t + cos1t + 1000p2 - tan t - tan1t + 999p2 - sin t + 4 sin1t - 1000p2
46. The height of the water, H, in feet, at a boat dock t hours after 6 A.M. is given by p H = 10 + 4 sin t. 6 a. Find the height of the water at the dock at 6 A.M., 9 A.M., noon, 6 P.M., midnight, and 3 A.M. b. When is low tide and when is high tide? c. What is the period of this function and what does this mean about the tides?
42. - cos t + 7 cos1t + 1000p2 + tan t + tan1t + 999p2 + sin t + sin1t - 1000p2
Writing in Mathematics
37. sin1t + 2p2 - cos1t + 4p2 + tan1t + p2 38. sin1t + 2p2 + cos1t + 4p2 - tan1t + p2 39. sin1 -t - 2p2 - cos1 -t - 4p2 - tan1 -t - p2 40. sin1 - t - 2p2 + cos1 - t - 4p2 - tan1 - t - p2
Application Exercises 43. The number of hours of daylight, H, on day t of any given year (on January 1, t = 1) in Fairbanks, Alaska, can be modeled by the function H1t2 = 12 + 8.3 sin c
2p 1t - 802 d . 365
a. March 21, the 80th day of the year, is the spring equinox. Find the number of hours of daylight in Fairbanks on this day. b. June 21, the 172nd day of the year, is the summer solstice, the day with the maximum number of hours of daylight. To the nearest tenth of an hour, find the number of hours of daylight in Fairbanks on this day.
47. Why are the trigonometric functions sometimes called circular functions? 48. What is the range of the sine function? Use the unit circle to explain where this range comes from. 49. What do we mean by even trigonometric functions? Which of the six functions fall into this category? 50. What is a periodic function? Why are the sine and cosine functions periodic? 51. Explain how you can use the function for emotional fluctuations in Exercise 45 to determine good days for having dinner with your moody boss. 52. Describe a phenomenon that repeats infinitely. What is its period?
Critical Thinking Exercises
c. December 21, the 355th day of the year, is the winter solstice, the day with the minimum number of hours of daylight. Find, to the nearest tenth of an hour, the number of hours of daylight in Fairbanks on this day.
Make Sense? In Exercises 53–56, determine whether each statement makes sense or does not make sense, and explain your reasoning.
44. The number of hours of daylight, H, on day t of any given year (on January 1, t = 1) in San Diego, California, can be modeled by the function
53. Assuming that the innermost circle on this Navajo sand painting is a unit circle, as A moves around the circle, its coordinates define the cosine and sine functions, respectively.
H1t2 = 12 + 2.4 sin c
2p 1t - 802 d . 365
a. March 21, the 80th day of the year, is the spring equinox. Find the number of hours of daylight in San Diego on this day.
A
b. June 21, the 172nd day of the year, is the summer solstice, the day with the maximum number of hours of daylight. Find, to the nearest tenth of an hour, the number of hours of daylight in San Diego on this day. c. December 21, the 355th day of the year, is the winter solstice, the day with the minimum number of hours of daylight. To the nearest tenth of an hour, find the number of hours of daylight in San Diego on this day. 45. People who believe in biorhythms claim that there are three cycles that rule our behavior—the physical, emotional, and mental. Each is a sine function of a certain period. The function for our emotional fluctuations is p E = sin t, 14 where t is measured in days starting at birth. Emotional fluctuations, E, are measured from - 1 to 1, inclusive, with
54. I’m using a value for t and a point on the unit circle corresponding to t for which sin t = -
110 . 2
p 23 p 13 = , I can conclude that cos a- b = . 6 2 6 2 7p 56. I can find the exact value of sin 3 using periodic properties of the sine function, or using a coterminal angle and a reference angle. 55. Because cos
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Mid-Chapter Check Point 57. Find the exact value of cos 0° + cos 1° + cos 2° + cos 3° + Á + cos 179° + cos 180°.
62. y = 4 sin A 2x x
58. If f1x2 = sin x and f1a2 = 14 , find the value of f1a2 + f1a + 2p2 + f1a + 4p2 + f1a + 6p2. 59. If f1x2 = sin x f1a2 + 2f1 - a2.
and
f1a2 = 14 ,
find
the
value
p 3
2p 3
535
B
7p 12
13p 12
5p 6
4p 3
y of
60. The seats of a Ferris wheel are 40 feet from the wheel’s center. When you get on the ride, your seat is 5 feet above the ground. How far above the ground are you after rotating through an angle of 765°? Round to the nearest foot.
63. y = 3 sin p2 x x
0
1 3
5 3
1
2
7 3
11 3
3
4
y
Preview Exercises Exercises 61–63 will help you prepare for the material covered in the next section. In each exercise, complete the table of coordinates. Do not use a calculator.
After completing this table of coordinates, plot the nine ordered pairs as points in a rectangular coordinate system. Then connect the points with a smooth curve.
61. y = 12 cos 14x + p2 x
- p4
- p8
0
p 8
p 4
y
Chapter
5
Mid-Chapter Check Point
What You Know: We learned to use radians to measure angles: One radian (approximately 57°) is the measure of the central angle that intercepts an arc equal in length to the radius of the circle. Using 180° = p radians, p b and we converted degrees to radians a multiply by 180° 180° b . We defined the six radians to degrees a multiply by p trigonometric functions using right triangles, angles in standard position, and coordinates of points along the unit circle. Evaluating trigonometric functions using reference angles involved connecting a number of concepts, including finding coterminal and reference angles, locating special angles, determining the signs of the trigonometric functions in specific quadrants, and finding the function values at special angles. Use the important Study Tip on page 524 as a reference sheet to help connect these concepts.
In Exercises 5–7, a. Find a positive angle less than 360° or 2p that is coterminal with the given angle. b. Draw the given angle in standard position. c. Find the reference angle for the given angle. 5.
11p 3
6. -
7. 510°
8. Use the triangle to find each of the six trigonometric functions of u. B
6
5
u A
C
9. Use the point on the terminal side of u to find each of the six trigonometric functions of u.
In Exercises 1–2, convert each angle in degrees to radians. Express your answer as a multiple of p. 1. 10°
19p 4
y
2. -105° u x
In Exercises 3–4, convert each angle in radians to degrees. 3.
5p 12
4. -
13p 20
P(3, ⫺2)
