Download MA 1165 - Practice Test III (Answers corrected) 1 4/13/09 (Test III is

MA 1165 - Practice Test III (Answers corrected)
4/13/09 (Test III is Wednesday, 4/15/09)
For problems 1-5, find all solutions between 0◦ and 360◦.
1.
sin(θ) = .75.
2.
sin(θ) = −.75.
3.
cos(θ) = .75.
4.
cos(θ) = −.75.
5.
cos(θ) =
√
3
2 .
For problems 6-9, use the law of cosines to find the required side.
6.
Given C = 20◦ , a = 15, and b = 3. Find c.
7.
Given b = 10, c = 3, and A = 72◦ . Find a.
8.
Given b = 7, c = 6, and A = 26◦ . Find a.
9.
Given a = 5, c = 5, and B = 45◦. Find b.
For problems 10-13, use the law of cosines to find the required angle.
10.
Given a = 3, b = 7, and c = 8. Find A.
11.
Given a = 73, b = 56, and c = 45. Find C.
12.
Given a = 10, b = 15, and c = 13. Find B.
13.
Given a = 8, b = 8, and c = 4. find B.
For problem 14 and 15, use the law of cosines to find the required side. Choose your answers from
14.
B = 35◦ , a = 6, and c = 4. Find b.
15.
B = 50◦ , a = 10, and c = 14. Find b.
For problem 16-25, use the law of sines.
16.
Given A = 32◦, B = 47◦, and a = 3, find b.
17.
Given A = 32◦, B = 47◦, and a = 3, find c.
18.
Given A = 72◦, C = 31◦, and c = 9, find a.
19.
Given A = 72◦, C = 31◦, and c = 9, find b.
20.
Given A = 34◦, a = 3, and c = 5, find C.
21.
From the previous problem, find B.
22.
Given C = 44◦ , b = 7, and c = 9, find B.
23.
From the previous problem, find A.
24.
Given a = 6, b = 7, and A = 44◦ . Find B.
25.
From the previous problem, find C.
26.
Write cot(θ) csc(θ) sin2 (θ) in terms of sines and cosines, and then simplify.
27.
Write
tan(θ) csc(θ)
sin2 (θ)
in terms of sines and cosines, and then simplify.
1
MA 1165 - Practice Test III (Answers corrected)
28.
29.
2
Write cos(θ) + sin2 (θ) in terms of cosines only.
Write 1 − cos2 (θ) sin2 (θ) in terms of sines only.
For problems 30-35, write with only θ inside the trig function.
3π
2 ).
30.
sin(θ +
31.
sin(θ − 90◦).
32.
cos(θ − 90◦ ).
33.
cos(θ − 450◦ ).
34.
cos(−θ + π2 ).
35.
cos( π2 − θ).
For problems 36-39, apply one of the addition formulas (equation group (5)) to find an exact value. Choose
your answers from
(a)
√ √
6+ 2
4
(b)
√ √
− 6+ 2
4
(c)
36.
cos(−15◦ ) = cos(30◦ − 45◦ ).
37.
sin(75◦ ) = sin(45◦ + 30◦).
38.
sin(105◦ ) = sin(135◦ − 30◦ ).
39.
cos(105◦) = cos(135◦ − 30◦ ).
√
√
6− 2
4
(d)
√ √
− 6− 2
4
(e) none of these
For problems 40-43, choose your answers from
(b) 4 cos3 (θ) − 3 cos(θ)
(d) 3 sin(θ) − 4 sin3 (θ)
(a) sin(θ) cos(2θ) + cos(θ) sin(2θ)
(c) cos(θ) cos(2θ) − sin(θ) sin(2θ)
40.
(e) none of these
Apply one of the addition formulas (group (5)) to cos(3θ) = cos(θ + 2θ), but do not simplify.
41.
Apply one of the double angle formulas (group (6)) to your expression from problem 5, and then use
sin2 (θ) = 1 − cos2 (θ) to get down to only cosines.
42.
Apply one of the addition formulas (group (5)) to sin(3θ) = sin(θ + 2θ).
43.
Apply one (or two) of the double angle formulas (group (6)) to your expression from problem 7, and
then use cos2 (θ) = 1 − sin2 (θ) to get down to only sines.
For problems 44 and 45, choose your answers from
(a)
1+cos(θ)
2
(b)
1−cos(6θ)
2
(c)
1+cos( 4θ )
2
(d)
1−cos( 3θ
2 )
2
(e) none of these
44.
Apply one of the power reduction formulas (group (7)) to sin2 (3θ).
45.
Apply one of the power reduction formulas (group (7)) to cos2 ( θ2 ).
For problems, 46-52 Simplify the following expressions.
46.
i6 .
47.
2i3 .
48.
−3i4 .
49.
(−i)5 .
50.
(5 − 2i)(7 + i).
MA 1165 - Practice Test III (Answers corrected)
51.
(1 − 3i)2 .
52.
(7i)(2 − 5i).
3
For problems 53-55, find the complex number that is r from the origin and makes an angle θ from the positive
real axis.
π
6.
53.
r = 2 and θ =
54.
r = 5 and θ = π.
55.
r = −2 and θ = 3π. (For negative r’s, go the opposite direction on the ray.)
Answers:
1) θ = 48.59◦, 131.41◦. 2) θ = 228.59◦, 311.41◦, 3) θ = 41.41◦, 318.59◦,
4) θ = 138.59◦, 221.41◦ (corrected), 5) θ = 30◦ , 330◦.
6) c = 12.2241. 7) a = 9.5110. 8) c = 3.0824. 9) b = 3.8268.
10) A = 21.79◦. 11) C = 38.03◦. 12) B = 80.26◦ 13) B = 75.52◦.
14) b = 3.5610. 15) b = 10.7712. 16) b = 4.1404. 17) c = 5.5572. 18) a = 16.6192.
19) b = 17.0266. 20) C = 68.75◦, 111.25◦ (corrected). 21) B = 77.25◦, 34.75◦ (corrected).
22) B = 32.70◦. 23) A = 103.30◦. 24) B = 54.14◦, 125.86◦. 25) C = 81.86◦, 10.14◦.
26) cos(θ). 27) cos(θ)1sin2 (θ) . 28) 1 + cos(θ) − cos2 (θ). 29) sin4 (θ).
30) − cos(θ). 31) − cos(θ). 32) sin(θ). 33) sin(θ). 34) sin(θ). 35) sin(θ).
√
√
√
√
√
√
√
√
2
2
36) 6+4 2 . 37) 6+
. 38) 6+4 2 . 39) − 6+
.
4
4
40) cos(θ) cos(2θ) − sin(θ) sin(2θ). 41) 4 cos3 (θ) − 3 cos(θ).
42) sin(θ) cos(2θ) + cos(θ) sin(2θ). 43) 3 sin(θ) − 4 sin3 (θ).
44)
1−cos(6θ)
.
2
45)
1+cos(θ)
.
2
46) −1. 47) −2i. 48) −3. 49) −i. 50) 37 − 9i. 51) −8 − 6i. 52) 35 + 14i. 53)
√
3 + i. 54) −5. 55) 2.
1 THE TRIGONOMETRIC IDENTITIES
1
4
The Trigonometric Identities
tan(θ) =
sin(θ)
cos(θ)
sec(θ) =
1
cos(θ)
(1)
cos(θ)
cot(θ) =
sin(θ)
1
csc(θ) =
sin(θ)
cos2 (θ) + sin2 (θ) = 1
1 + tan2 (θ) = sec2 (θ)
(2)
1 + cot2 (θ) = csc2 (θ)
sin(−θ) = − sin(θ).
(3)
cos(−θ) = cos(θ).
in radians
in degrees
sin(θ + 2π) = sin(θ)
sin(θ ± 360◦ ) = sin(θ)
cos(θ + 2π) = cos(θ)
cos(θ ± 360◦) = cos(θ)
sin(θ ± π) = − sin(θ)
sin(θ ± 180◦ ) = − sin(θ)
cos(θ ± π) = − cos(θ)
cos(θ ± 180◦) = − cos(θ)
sin(θ + π2 ) = cos(θ)
sin(θ + 90◦ ) = cos(θ)
cos(θ + π2 ) = − sin(θ)
cos(θ + 90◦) = − sin(θ)
(4)
sin(α + β) = sin(α) cos(β) + cos(α) sin(β)
sin(α − β) = sin(α) cos(β) − cos(α) sin(β)
(5)
cos(α + β) = cos(α) cos(β) − sin(α) sin(β)
cos(α − β) = cos(α) cos(β) + sin(α) sin(β)
sin(2θ) = 2 sin(θ) cos(θ)
sin2 (θ) =
1 − cos(2θ)
2
and
and
cos(2θ) = cos2 (θ) − sin2 (θ)
cos(2θ) = 2 cos2 (θ) − 1
cos(2θ) = 1 − 2 sin2 (θ)
cos2 (θ) =
1 + cos(2θ)
2
(6)
(7)