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Math 1316 Exam 3 Review (Chapters 5 & 6)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Use the fundamental identities to find the value of the trigonometric function.
1) Find sin θ if cot θ = - 2 and cos θ < 0.
1)
Perform the transformation.
2) Write sec x in terms of sin x.
2)
Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
csc x cot x
3)
3)
sec x
4) sin2 x + sin2 x cot2 x
4)
Perform the indicated operations and simplify the result so there are no quotients.
sin θ
sin θ
5)
- 1 + sin θ 1 - sin θ
6)
(sin θ + cos θ)2
1 + 2 sin θ cos θ
5)
6)
Use the fundamental identities to simplify the expression.
7) cos (-x) cos x - sin (-x) sin x
7)
Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
sin2x - 1
8)
8)
cos (-x)
Use the fundamental identities to simplify the expression.
9) sin2θ + tan2θ + cos2θ
9)
Verify that each equation is an identity.
10) (1 + tan2s)(1 - sin2s) = 1
10)
11)
sec θ - 1
tan θ
= tan θ
sec θ + 1
11)
Use the fundamental identities to simplify the expression.
sin x cos x
12)
tan x
Use Identities to find the exact value.
13) cos (-75°)
12)
13)
1
14) cos 7π
5π
7π
5π
cos + sin sin 12
12
12
12
14)
15) cos π
12
15)
Find the exact value of the expression using the provided information.
1
1
16) Find cos(s + t) given that cos s = , with s in quadrant I, and sin t = - , with t in
3
2
16)
quadrant IV.
Use a sum or difference identity to find the exact value.
17) sin 25° cos 35° + cos 25° sin 35°
18)
17)
tan 80° + tan 70°
1 - tan 80° tan 70°
18)
Find the exact value of the expression using the provided information.
1
3
19) Find sin(s + t) given that cos s = - , with s in quadrant III, and cos t = - , with t in
4
5
19)
quadrant III.
20) Find tan(s - t) given that sin s = - 3 13
10
, with s in quadrant IV, and sin t = - , with
13
10
20)
t in quadrant IV.
Use a sum or difference identity to find the exact value.
21) tan 105°
21)
Using a sum or difference identity, write the following as an expression involving functions of x.
π
22) sin x - 22)
2
23) tan (30° + x)
23)
Use an identity to write the expression as a single trigonometric function or as a single number.
24) sin 22.5° cos 22.5°
24)
2 tan 15°
1 - tan2 15°
25)
26) 2 cos24x - 1
26)
27) 4 sin 2x cos 2x
27)
25)
2
28) cos24x - sin24x
28)
Use identities to find the indicated value for each angle measure.
4
29) cos 2θ = and θ terminates in quadrant III
Find cos θ.
5
Find the exact value by using a half-angle identity.
30) tan 75°
29)
30)
31) cos 22.5°
31)
Use identities to find the indicated value for each angle measure.
12
3π
32) tan θ = , π < θ < Find sin(2θ).
5
2
Determine all solutions of the equation in radians.
x
1
π
33) Find cos , given that cos x = and x terminates in 0 < x < .
2
4
2
x
34) Find tan , given that tan x = -3 and x terminates in 90° < x < 180°.
2
32)
33)
34)
Use an identity to write the expression as a single trigonometric function or as a single number.
1 + cos 26°
35)
35)
2
Find the exact value of the real number y.
36) y = arctan 1
36)
3
37) y = cos-1 2
37)
Use a calculator to give the real number value. Round the answer to 7 decimal places.
38) y = arcsec (2.8842912)
38)
Graph the inverse circular function.
39) y = sin-1 x
39)
Give the exact value of the expression.
1
40) cos arcsin 4
41) arccos cos 40)
4π
3
41)
3
Write the following as an algebraic expression in u, u > 0.
u
42) sin arctan 5
43) tan cos-1 42)
u
3
43)
Solve the equation for exact solutions over the interval [0, 2π).
44) cos2x + 2 cos x + 1 = 0
45) 2 sin2x = sin x
44)
45)
Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary.
46) 4 sin2θ = 3
46)
47) 3 sin2θ - sin θ - 4 = 0
47)
48) 2 cos2θ + 7 sin θ = 5
48)
Solve the equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate
answers in radians to four decimal places and approximate answers in degrees to the nearest tenth.
49) 2 sin2 x + sin x = 1
49)
Solve the equation for solutions in the interval [0, 2π).
3
50) sin 4x = 2
50)
51) 2 cos 2x = 1
51)
Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree.
θ
52) cot = 1
3
53) tan2 2θ = 5
52)
53)
Solve the equation for x, where x is restricted to the given interval.
π π
54) y = 7 sin x, for x in - , 2 2
Solve the equation for exact solutions.
π
55) -sin-1(4x) = 4
54)
55)
4
Answer Key
Testname: UNTITLED1
1)
5
5
2)
± 1 - sin2 x
1 - sin2 x
3) cot2 x
4) 1
5) -2 tan2θ
6) 1
7) 1
8) -cos x
9) sec2θ
10) (1 + tan2s)(1 - sin2s) = sec2s · cos2s = 11)
1
cos2s
· cos2s = 1
sec θ - 1 sec θ - 1 sec θ + 1
sec2 θ - 1
tan2 θ
tan θ
= · = = = tan θ
tan θ
sec θ + 1 tan θ(sec θ + 1) tan θ(sec θ + 1) sec θ + 1
12) cos2 x
6 - 2
13)
4
14)
3
2
15)
6 + 2
4
16)
3 + 2 2
6
17)
3
2
18) - 19)
3
3
3 15 + 4
20
20) - 7
9
21) -2 - 3
22) -cos x
1 + 3 tan x
23)
3 - tan x
24)
2
4
25)
3
3
26) cos 8x
27) 2 sin 4x
28) cos 8x
5
Answer Key
Testname: UNTITLED1
29) cos θ = - 3 10
10
30) 2 + 3
1
31) 2 + 2
2
32)
120
169
33)
10
4
34)
10 + 1
3
35) cos 13°
π
36)
4
37)
π
6
38) 1.2167397
39)
y


2
-1
1
x
-
2
-
40)
15
4
41)
2π
3
42)
u u2 + 5
u2 + 5
43)
9 - u2
u
44) {π}
π 5π
45) 0, π, , 6 6
46) {60°, 120°, 240°, 300°}
47) {270°}
48) {90°, 48.6°, 131.4°}
6
Answer Key
Testname: UNTITLED1
49)
π
5π
3π
+ 2nπ, + 2nπ, + 2nπ
6
6
2
50)
π π 2π 7π 7π 13π 5π 19π
, , , , , , , 12 6 3 12 6
12 3
12
51)
π 9π 7π 15π
, , , 8 8
8
8
52) {135°}
53) {33°, 57°, 123°, 147°, 213°, 237°, 303°, 327°}
y
54) x = arcsin 7
55) - 2
8
7