Download Section 6.1

Jim Lambers
Math 1B
Fall Quarter 2004-05
Quiz 6 Practice Problems
The following problems are indicative of the types of problems that will appear on Quiz 6, which
will be given in class on Thursday, November 18. The exercises are from the textbook, Precalculus,
5th Edition, by Barnett, Ziegler, and Byleen. As they are odd-numbered exercises, you can find
the answers at the end of this document.
Section 6.1
1-63 Verify the following identities.
1. sin θ sec θ = tan θ
3. cot u sec u sin u = 1
sin(−x)
5. cos(−x)
= − tan x
α cot α
7. sin α = tancsc
α
9. cot u + 1 = (csc u)(cos u + sin u)
x−sin x
11. cos
sin x cos x = csc x − sec x
13.
15.
sin2 t
cos t + cos t = sec t
cos x
= sec x
1−sin2 x
17.
19.
21.
23.
37.
(1 − cos u)(1 + cos u) = sin2 u
cos2 x − sin2 x = 1 − 2 sin2 x
(sec t + 1)(sec t − 1) = tan2 t
csc2 x − cot2 x = 1
1−(sin x−cos x)2
= 2 cos x
sin x
cot θ+1
csc θ
1+cos y
sin2 y
1−cos y = (1−cos y)2
tan2 x − sin2 x = tan2 x sin2 x
39. cos θ + sin θ =
41.
43.
45.
51.
csc θ
cot θ+tan θ = cos θ
1−cos A
sec A−1
1+cos A = sec A+1
sin4 w − cos4 w =
53.
1 − 2 cos2 w
x
59. (sec x − tan x)2 = 1−sin
1+sin x
63.
1+sin v
cos v
=
cos v
1−sin v
1
Section 6.2
11-13 Verify each identity using cofunction identities for sine and cosine, and basic identities discussed in Section 6.1.
11. cot π2 − x = tan x
13. csc π2 − x = sec x
15-19 Convert each expression to forms involving sin x, cos x, and/or tan x using sum or difference
identities.
15. cos(x + 45◦ )
17. tan π3 + x
19. sin(x − 90◦ )
21-27 Use appropriate identities to find exact values. Do not use a calculator.
21. cos 20◦ cos 25◦ − sin 20◦ sin 25◦
23.
25.
tan 50◦ −tan 20◦
1+tan 50◦ tan 20◦
sin 15◦
27. cos 11π
12 Hint:
11π
12
=
2π
3
+
π
4
29-31 Find sin(x − y) and tan(x + y) exactly without a calculator using the information given.
√
29. sin x = −3/5, sin y = 8/3, x is a Quadrant IV angle, y is a Quadrant I angle
31. tan x = 3/4, tan y = −1/2, x is a Quadrant III angle, y is a Quadrant IV angle
33-43 Verify each identity.
33. cos 2x = cos2 x − sin2 x
cot x cot y−1
cot x+cot y
2 tan x
1−tan2 x
35. cot(x + y) =
37. tan 2x =
39.
sin(v+u)
sin(v−u)
=
cot u+cot v
cot u−cot v
cos(x+y)
sin x cos y
cot y−cot x
cot x cot y+1
41. cot x − tan y =
43. tan(x − y) =
2
Section 6.3
7-9 Find the exact value without a calculator using half-angle identities.
7. tan 15◦
9. cos 112.5◦
15-27 Verify each identity.
15. (sin x + cos x)2 = 1 + sin 2x
17. sin2 x = 21 (1 − cos 2x)
19. 1 − cos 2x = tan x sin 2x
x
1−cos x
2 =
2
sin θ
θ
2 = 1−cos θ
21. sin2
23. cot
25. cos 2u =
1−tan2 u
1+tan2 u
27. 2 csc 2x =
1+tan2 x
tan x
35-37 Compute the exact values sin 2x, cos 2x, and tan 2x using the information given and appropriate identities. Do not use a calculator.
35. sin x = 3/35, π/2 < x < π
37. tan x = −5/12, −π/2 < x < 0
39-41 Compute the exact values of sin(x/2), cos(x/2) and tan(x/2) using the information given and
appropriate identities. Do not use a calculator.
39. sin x = −1/3, π < x < 3π/2
41. cot x = 3/4, −π < x < −π/2
Answers
Section 6.1
1. Use sec θ = 1/ cos θ and tan θ = sin θ/ cos θ
3. Use cot u = cos u/ sin u and sec u = 1/ cos u
5. Use sin(−x) = − sin x, cos(−x) = cos x, and tan x = sin x/ cos x
7. Use tan α = 1/ cot α and csc α = 1/ sin α
3
9. Use csc u = 1/ sin u and cot u = cos u/ sin u
11. Use csc x = 1/ sin x and sec x = 1/ cos x
13. Use sin2 t + cos2 t = 1 and sec t = 1/ cos t
15. Use sin2 x + cos2 x = 1 and sec x = 1/ cos x
17. Use sin2 u + cos2 u = 1
19. Use sin2 x + cos2 x = 1
21. Use sec t = 1/ cos t, tan t = sin t/ cos t, and sin2 t + cos2 t = 1
23. Use csc x = 1/ sin x, cot x = cos x/ sin x, and sin2 x + cos2 x = 1
37. Use sin2 x + cos2 x = 1
39. Use cot θ = cos θ/ sin θ and csc θ = 1/ sin θ
41. Use sin2 y + cos2 y = 1
43. Use sin2 x + cos2 x = 1
45. Use cot θ = cos θ/ sin θ, tan θ = sin θ/ cos θ, sin2 θ + cos2 θ = 1, and csc θ = 1/ sin θ
51. Use sec A = 1/ cos A
53. Use sin2 w + cos2 w = 1
59. Use sec x = 1/ cos x and tan x = sin x/ cos x
63. Use sin2 v + cos2 v = 1
Section 6.2
√
15.
17.
2
2 (cos x −
√
3+tan
x
√
1− 3 tan x
sin x)
19. − cos x
√
21.
23.
2
2
√
3
3
4
√
25.
27.
√
2
4 ( 3 − 1)
√
√
− 42 (1 + 3)
29. sin(x − y) =
√
−3−4 8
,
15
31. sin(x − y) =
−2
√
,
5
tan(x + y) =
√
4 8−3
√
4+3 8
tan(x + y) = 2/11
Section 6.3
√
7. 2 − 3
√ √
2
9. − 2−
2
24
, cos 2x =
35. sin 2x = − 25
7
25 ,
tan 2x = − 24
7
120
120
, cos 2x = 119
37. sin 2x = − 169
169 , tan 2x = − 119
q √
q √
√
39. sin x2 = 3+26 2 , cos x2 = − 3−26 2 , tan x2 = −3 − 2 2
√
41. sin x2 = − 2 5 5 , cos x2 =
√
5
5 ,
tan x2 = −2
5