5.3
Sum and Difference Identities for Cosine
207
In Example 4, the values of cos s and sin t could also be found by using
the Pythagorean identities. The problem could then be solved using the identity
for cos(s + t) in the same way as shown in the example.
NOT
• ••
E
Example 5
Applying the Cosine Difference Identity to Voltage
Common household electrical current is called alternating current because the current alternates direction within the wires. The voltage V
~
in a typical lIS-volt outlet can be expressed using the equation V =
163 sin wt, where w is the angular velocity (in radians per second) of the rotating generator at the electrical plant, and t is time measured in seconds. (Source:
Bell, D., Fundamentals of Electric Circuits, Fourth Edition, Prentice-Hall, 1988.)
e
(a) It is essential for electrical generators to rotate at precisely 60 cycles per second so household appliances and computers will function properly. Determine w for these electrical generators.
Since each cycle is 27T radians, at 60 cycles per second, w = 60(27T) =
1207T radians per second.
(b) Graph Von the interval 0 ::5 t ::5 .05.
V = 163 sin wt = 163 sin 120m. Because the amplitude is 163 here, we
choose -200 ::5 V::5 200 for the range, as shown in Figure 10.
o 1F---I----+.0...4----,L-....+--:t
(e) For what value of qy will the graph of V = 163 cos(wt - qy) be the same as
the graph of V = 163 sin wt?
Since cos(x - 7T/2) = cos( 7T/2 - x) = sin x, choose qy = 7T/2. In the
Chapter Test you will be asked to use the cosine difference identity to show that
V = 163 sin wt and V = 163 cos(wt - 7T/2) are equivalent equations.
I
-200
•••
Figure 10
5.3
Exercises
+ B). How do they differ? How are they alike?
~
1. Compare the formulas for cos(A - B) and cos(A
~
2. What does the cofunction identity cos( 7T/2 - (J) = sin (J imply about the graphs of the cosine and sine functions? (Hint:
First observe that cos(7T/2 - (J) is the same as cos((J - 7T/2).)
Use the cosine sum and difference identities to find each exact value. (Do not use a calculator.) See Example 1.
3. cas 75°
5. cas 105°
4. cos( -15°)
(Hint: 105° = 60°
+ 45°)
6. cos( -105°)
(Hint: -105° = -60°
+ (-45°))
77T
7. cas
12
9. cas 40° cas 50° - sin 40° sin 50°
27T
11. cos-cos5
~
7T
27T.
- sin-sm10
5
7T
10
10. cos( -10°) cas 35°
+ sine-10°) sin 35°
77T
27T
77T 27T
12. cas - cas - - sin - sin 9
9
9
9
Use a graphing calculator to support your answer for each of the following. See Example 1.
13. Exercise 9
14. Exercise 10
208
Chapter 5
Write each function
Trigonometric Identities
value in terms of the cofunction
of a complementary
angle. See Example
2.
15. tan 87°
16. sin 15°
17. cos-
7r
12
27r
18. sin5
19. csc( -14° 24')
20. sin 142° 14'
57r
21. sin8
97r
22. cot10
23. see 146° 42'
24. tan 174° 3'
25. cot 176.9814°
26. sin 98.0142°
Use the cofunction
27. cot-
identities to fill in each blank with the appropriate
7r
=
3
7r
6
_
30. ___
28. sin 2; =
72° = cot 18°
Use the cofunction
31. cos 70° = ----___
+
35. see e = csc (~
+ 200)
+ 25°)
43. cos(O°
+
e)
Find cos(s
+
t) and cos(s - t). See Example 4.
44. cos(90°
50. cos s = -8/17
and cos t = -3/5,
+
as a single function
41. cos(180° - e)
45. cos(180°
e)
+
46. cos(270°
e)
+ sin 60° sin 14°
56. cos 140° = cos 60° cos 80° - sin 60° sin 80°
27r
117r
7r
58. cos - = cos -- cos 3
12
4
60. cos 85° cos 40°
= cot
e
62. sin (
e - ;)
= -sinx
65. cos 2x = cos" X
+ cos 2x
-
sin?x
64. secl-rr (Hint: cos 2x = cos(x
- cos? X = cos? x
+ x).)
(Hint: Use the result from Exercise 65.)
x) =
117r
tr
sin 12
4
+ sin --
+ sin 85° sin 400 = v'2
= cos
Verify that each equation is an identity.
~. 66. 1
e)
54. cos( -24°) = cos 16° - cos 40°
59. cos 70° cos 20° - sin 70° sin 20° = 0
x)
+
is true or false.
7r
7r
.7r.7r
= cos - cos - - sm - sm 12
4
12
4
+
of e. See Example 3.
42. cos(270° - e)
s and t in quadrant IV
+ 12°)
55. cos 74° = cos 60° cos 14°
63. cos(;
+ 20°)
s and t in quadrant III
Tell whether each statement
e - ;)
+ 10°)
and sin t = \16/8, s and t in quadrant I
53. cos 42°'= cos(30°
61. tan (
true. See Example 2.
s in quadrant I and t in quadrant III
52. cos s = v'2/4 and sin t = -Vs/6,
3
66°
s in quadrant II and t in quadrant IV
49. sin s = 3/5 and sin t = -12/13,
-tt
33° = sin 57°
and sin t = 3/5, sand t in quadrant II
48. sin s = 2/3 and sin t = -1/3,
57. cos -
32. tan 24° = ----___
to write each expression
40. cos(90° - e)
Check
20°
1
38. cot(e - 10°) = tan(2e
Use the identities for the cosine of a sum or a difference
Concept
29. ___
36. cos e = sin(3 e
39. cos(O° - e)
51. sin s = Vs/7
name. See Example 2.
34. sin e = cos(2e - 109)
2e)
37. sin(3e - 15°) = cos(e
function
( _-; )
identities to find an angle e that makes each statement
33. tan e = cot(45°
47. cos s = -1/5
trigonometric
-secx
2
e