406
CHAPTER 5
Trigonometric Functions of Real Numbers
x-coordinate is positive and its y-coordinate is negative. Thus, the desired
terminal point is
a
12
12
,
b
2
2
(c) The reference number is t p/3, which determines the terminal point
A 12, 13/2B from Table 1. Since the terminal point determined by t is in quadrant
III, its coordinates are both negative. Thus, the desired terminal point is
1
13
a , b
2
2
■
Since the circumference of the unit circle is 2p, the terminal point determined by
t is the same as that determined by t 2p or t 2p. In general, we can add or subtract 2p any number of times without changing the terminal point determined by t.
We use this observation in the next example to find terminal points for large t.
Example 7
y
Finding the Terminal Point for Large t
Find the terminal point determined by t 3 1
!_ Ϸ
, @
2 2
1
0
Solution Since
x
t
5.1
■
Exercises
Show that the point is on the unit circle.
4
3
1. a , b
5
5
2. a
5
2 16
b
4. a , 7
7
15 2
, b
5. a
3 3
5 12
,
b
13 13
■
3. a
7 24
,
b
25 25
111 5
, b
6. a
6
6
Find the missing coordinate of P, using the fact that P
7–12
lies on the unit circle in the given quadrant.
Coordinates
7.
Quadrant
B
PA 35,
257 B
8. PA
,
9. PA
, 13 B
10.
PA 25,
11. PA
12.
PA 23,
29p
5p
4p 6
6
we see that the terminal point of t is the same as that of 5p/6 (that is, we subtract
4p). So by Example 6(a) the terminal point is A13/2, 12 B . (See Figure 10.)
Figure 10
1–6
29p
.
6
III
IV
II
B
I
, 27 B
B
IV
II
13–18 ■ The point P is on the unit circle. Find P1x, y 2 from
the given information.
13. The x-coordinate of P is 45 and the y-coordinate is
positive.
14. The y-coordinate of P is 13 and the x-coordinate is
positive.
15. The y-coordinate of P is 23 and the x-coordinate is
negative.
16. The x-coordinate of P is positive and the y-coordinate of
P is 15/5.
17. The x-coordinate of P is 12/3 and P lies below the
x-axis.
18. The x-coordinate of P is 25 and P lies above the
x-axis.
■
SECTION 5.1
19–20 ■ Find t and the terminal point determined by t for each
point in the figure. In Exercise 19, t increases in increments of
p/4; in Exercise 20, t increases in increments of p/6.
19.
20.
y
π
t= 4 ;
1
1
2 Ϸ2
! Ϸ
, 2 @
2
1
_1
x
π
t= 6 ;
3 1
! Ϸ
, @
2 2
1
_1
_1
5p
7
(c) t 3
(b) t 11p
5
(c) t 6
(b) t 35. (a) t x
7p
9
9p
7
(d) t 7
37–50 ■ Find (a) the reference number for each value of t,
and (b) the terminal point determined by t.
37. t 2p
3
38. t 4p
3
39. t 3p
4
40. t 7p
3
_1
21–30 ■ Find the terminal point P1x, y 2 on the unit circle
determined by the given value of t.
41. t 2p
3
42. t 7p
6
21. t p
2
22. t 3p
2
43. t 13p
4
44. t 13p
6
23. t 5p
6
24. t 7p
6
45. t 7p
6
46. t 17p
4
26. t 5p
3
47. t 48. t 31p
6
25. t 27. t p
3
2p
3
29. t 28. t 3p
4
30. t p
2
11p
6
31. Suppose that the terminal point determined by t is the point
A 35, 45 B on the unit circle. Find the terminal point determined
by each of the following.
(a) p t
(b) t
(c) p t
(d) 2p t
32. Suppose that the terminal point determined by t is the point
A 34, 17/4B on the unit circle. Find the terminal point determined by each of the following.
(a) t
(b) 4p t
(c) p t
(d) t p
33–36
■
Find the reference number for each value of t.
5p
33. (a) t 4
p
6
34. (a) t 5p
6
(b) t 7p
6
(c) t 11p
3
(d) t 4p
3
49. t 11p
3
16p
3
50. t 41p
4
51–54 ■ Use the figure to find the terminal point determined
by the real number t, with coordinates correct to one decimal
place.
51. t 1
y
52. t 2.5
2
1
53. t 1.1
54. t 4.2
3
0
_1
1
6
x
4
7p
(b) t 3
(d) t (c) t 407
(d) t 5
36. (a) t y
The Unit Circle
5
Discovery • Discussion
7p
4
55. Finding the Terminal Point for P/6 Suppose the terminal point determined by t p/6 is P1x, y 2 and the points Q
and R are as shown in the figure on the next page. Why are
the distances PQ and PR the same? Use this fact, together
with the Distance Formula, to show that the coordinates of
416
CHAPTER 5
5.2
Trigonometric Functions of Real Numbers
Exercises
1–2 ■ Find sin t and cos t for the values of t whose terminal
points are shown on the unit circle in the figure. In Exercise 1,
t increases in increments of p/4; in Exercise 2, t increases in
increments of p/6. (See Exercises 19 and 20 in Section 5.1.)
1.
2.
y
y
1
1
7p
6
(c) tan
11p
6
16. (a) cot a
p
b
3
(b) cot
2p
3
(c) cot
5p
3
17. (a) cos a
p
b
4
(b) csc a
18. (a) sin
π
t= 6
1
_1
x
x
_1
_1
3–22 ■ Find the exact value of the trigonometric function at the
given real number.
2p
3
(b) cos
2p
3
(c) tan
2p
3
4. (a) sin
5p
6
(b) cos
5p
6
(c) tan
5p
6
5. (a) sin
7p
6
(b) sin a
(c) sin
11p
6
p
b
6
5p
(b) cos a
b
3
7p
(c) cos
3
3p
7. (a) cos
4
5p
(b) cos
4
7p
(c) cos
4
3p
4
(b) sin
5p
4
(c) sin
7p
4
9. (a) sin
7p
3
(b) csc
7p
3
(c) cot
7p
3
19. (a) csc a
p
b
2
(b) sec
5p
4
(c) tan
5p
4
(b) csc
p
2
(c) csc
3p
2
(b) sec p
(c) sec 4p
21. (a) sin 13p
(b) cos 14p
(c) tan 15p
22. (a) sin
25p
2
(b) cos
25p
2
(c) cot
24. t t
sin t
cos t
0
0
1
p
2
tan t
25. t p
csc t
p
0
undefined
3p
2
27–36 ■ The terminal point P1x, y 2 determined by a real
number t is given. Find sin t, cos t, and tan t.
(c) tan a
p
b
3
11. (a) sin a
p
b
2
(b) cos a
p
b
2
(c) cot a
p
b
2
29. a
12. (a) sin a
3p
b
2
(b) cos a
3p
b
2
(c) cot a
3p
b
2
6 113
b
31. a ,
7 7
32. a
40 9
,
b
41 41
p
b
3
33. a
5
12
, b
13
13
34. a
15 2 15
,
b
5
5
35. a
20 21
,
b
29 29
36. a
7
24
, b
25
25
11p
3
(c) sec a
14. (a) cos
7p
6
(b) sec
7p
6
(c) csc
cot t
p
2
p
b
3
(b) csc
7p
6
3p
2
undefined
(b) sec a
11p
3
25p
2
26. t sec t
p
b
3
13. (a) sec
p
b
4
20. (a) sec1p2
23. t 0
5p
6. (a) cos
3
8. (a) sin
5p
4
(c) cot a
p
b
4
23–26 ■ Find the value of each of the six trigonometric
functions (if it is defined) at the given real number t. Use your
answers to complete the table.
3. (a) sin
10. (a) cos a
(b) tan
1
π
t= 4
_1
5p
6
15. (a) tan
3 4
27. a , b
5 5
15
111
,
b
4
4
3 4
28. a , b
5 5
1
212
b
30. a , 3
3
SECTION 5.2
Trigonometric Functions of Real Numbers
37–44 ■ Find the approximate value of the given trigonometric
function by using (a) the figure and (b) a calculator. Compare
the two values.
y
37. sin 1
2
38. cos 0.8
1
71–78 ■ Determine whether the function is even, odd,
or neither.
39. sin 1.2
77. f 1x 2 x 3 cos x
71. f 1x 2 x 2 sin x
72. f 1x 2 x 2 cos 2x
75. f 1x 2 0 x 0 cos x
76. f 1x 2 x sin3x
73. f 1x 2 sin x cos x
417
74. f 1x 2 sin x cos x
78. f 1x 2 cos1sin x2
40. cos 5
41. tan 0.8
3
42. tan(1.3)
0
_1
1
6
43. cos 4.1
x
44. sin(5.2)
4
5
Applications
79. Harmonic Motion The displacement from equilibrium
of an oscillating mass attached to a spring is given by
y1t2 4 cos 3pt where y is measured in inches and t in
seconds. Find the displacement at the times indicated in
the table.
t
45–48 ■ Find the sign of the expression if the terminal point
determined by t is in the given quadrant.
45. sin t cos t,
47.
tan t sin t
,
cot t
quadrant II
46. tan t sec t,
quadrant IV
quadrant III
48. cos t sec t,
any quadrant
49–52 ■ From the information given, find the quadrant in
which the terminal point determined by t lies.
49. sin t 0 and cos t 0
50. tan t 0 and sin t 0
51. csc t 0 and sec t 0
52. cos t 0 and cot t 0
53–62 ■ Write the first expression in terms of the second if the
terminal point determined by t is in the given quadrant.
53. sin t, cos t ;
quadrant II
54. cos t, sin t;
quadrant IV
55. tan t, sin t ;
quadrant IV
56. tan t, cos t;
quadrant III
57. sec t, tan t ;
quadrant II
58. csc t, cot t;
quadrant III
59. tan t, sec t;
quadrant III
60. sin t, sec t ;
quadrant IV
2
61. tan t, sin t ;
y>0
Equilibrium, y=0
y<0
80. Circadian Rhythms Everybody’s blood pressure
varies over the course of the day. In a certain individual
the resting diastolic blood pressure at time t is given by
B1t2 80 7 sin1pt/12 2 , where t is measured in hours
since midnight and B1t2 in mmHg (millimeters of mercury).
Find this person’s diastolic blood pressure at
(a) 6:00 A.M. (b) 10:30 A.M. (c) Noon (d) 8:00 P.M.
81. Electric Circuit After the switch is closed in the circuit
shown, the current t seconds later is I1t2 0.8e 3tsin 10t.
Find the current at the times
(a) t 0.1 s and (b) t 0.5 s.
L
any quadrant
63–70 ■ Find the values of the trigonometric functions of t
from the given information.
64. cos t 0
0.25
0.50
0.75
1.00
1.25
any quadrant
62. sec2t sin2t, cos t ;
63. sin t 35,
y1t2
R
C
terminal point of t is in quadrant II
45,
terminal point of t is in quadrant III
65. sec t 3,
terminal point of t is in quadrant IV
66. tan t 14,
terminal point of t is in quadrant III
67. tan t 34,
cos t 0
68. sec t 2,
69. sin t 14,
sec t 0
70. tan t 4,
sin t 0
csc t 0
E
L 103 h
R 6 103 C 9.17 mf
E 4.8 103 V
S
82. Bungee Jumping A bungee jumper plummets from
a high bridge to the river below and then bounces back
over and over again. At time t seconds after her jump,
her height H (in meters) above the river is given by