Trigonometry
Exam 2 Review
6.4, 6.5, 6.6 – Graphing trig functions:
1. Use the amplitude, period, and phase shift to graph one period of each functions.
a. 𝑦 = 3 sin 4𝑥
b. 𝑦 = −2 cos 2𝑥
3
e. 𝑦 = −3 cos(𝑥 + 𝜋)
2
2. Graph two periods of each function.
1
𝜋
e. 𝑦 = − cot 𝑥
2
2
𝜋
f. 𝑦 = cos �2𝑥 + �
h. 𝑦 = sin(2𝑥) + 1
a. 𝑦 = 4 tan 𝑥
c. 𝑦 = 3 cos
4
𝜋
d. 𝑦 = − sin 𝜋𝑥
3
𝜋
g. 𝑦 = −3 sin � 𝑥 − 3𝜋�
3
𝜋
b. 𝑦 = −2 tan 𝑥
i. 𝑦 = 3 sec(𝑥 + 𝜋)
𝑥
4
c. 𝑦 = − tan �𝑥 − �
𝜋
f. 𝑦 = 2 cot �𝑥 + �
2
5
4
j. 𝑦 = csc(𝑥 − 𝜋)
g. 𝑦 = 3 sec 2𝜋𝑥
d. 𝑦 = 2 cot 3𝑥
h. 𝑦 = −2 csc 𝜋𝑥
2
7.1, 7.2 – Inverse trig functions:
3.
4.
Find the exact value of each expression.
a. sin−1 1
c. tan−1 1
d. sin−1 �−
Find the exact value of each expression.
a. cos �sin−1
5.
b. cos−1 1
√2
2
�
4
5
e. tan �cos−1 �− ��
√3
�
2
1
2
e. cos −1 �− �
3
b. sin(cos−1 0)
1
3
3
4
c. cos �tan−1 4�
f. sin �tan−1 �− ��
f. tan−1 �−
d. tan �sin−1 �− ��
Find the exact value of each expression or write undefined if necessary.
a. sin �sin−1
√7
8
i. sin−1 �sin
2𝜋
�
3
�
e. cos(cos −1 𝜋)
b. sin �sin−1
1
8
√7
f. tan �tan−1 2�
j. sin−1 �sin
7𝜋
�
9
�
3
c. cos �cos−1 4�
g. tan(tan−1 2)
𝜋
4
k. cos−1 �cos �− ��
d. cos(cos−1 3.14)
𝜋
7
h. sin−1 �sin �
l. cos −1 �cos �−
27𝜋
��
14
√3
�
3
7.3 – Trig equations:
6.
7.
For each equation, find all solutions.
a. cos 𝑥 = −
1
2
b. sin 𝑥 =
√2
2
c. 2 sin 𝑥 + 1 = 0
Solve the equation on [0,2𝜋).
a. cos 2𝑥 = −1
b. 4 sin 3𝑥 − 4 = 0
i. sin 𝑥 = tan 𝑥
j. sin 𝑥 = −0.6031
e. cos2 𝑥 − 2cos 𝑥 = 3
𝑥
2
d. √3 tan 𝑥 − 1 = 0
c. tan = −1
f. 2 cos 2 𝑥 − sin 𝑥 = 1
e. cos 2𝑥 = −1
d. tan 𝑥 = 2 cos 𝑥 tan 𝑥
g. 4 sin2 𝑥 = 1
k. 5 cos2 𝑥 − 3 = 0
h. cos2 𝑥 − sin2 𝑥 − sin 𝑥 = 1
l. sec 2 𝑥 = 4 tan 𝑥 − 2
7.4 – Verifying identities:
8.
Verify each identity.
a. sec 𝑥 − cos 𝑥 = tan 𝑥 sin 𝑥
d. (sec 𝜃 − 1)(sec 𝜃 + 1) = tan2 𝜃
g.
1+sin 𝑡
cos2 𝑡
= tan2 𝑡 + 1 + tan 𝑡 sec 𝑡
j. (tan 𝜃 + cot 𝜃)2 = sec 2 𝜃 + csc 2 𝜃
l.
cos 𝑡
cot 𝑡−5 cos 𝑡
=
1
csc 𝑡−5
m.
c. sin2 𝜃 (1 + cot 2 𝜃) = 1
b. cos 𝑥 + sin 𝑥 tan 𝑥 = sec 𝑥
1−cos 𝑡
1+cos 𝑡
e.
h.
1−tan 𝑥
sin 𝑥
cos 𝑥
1−sin 𝑥
k.
f.
= csc 𝑥 − sec 𝑥
=
1+sin 𝑥
cos 𝑥
1
sin 𝜃+𝑐𝑜𝑠𝜃
+
= (csc 𝑡 − cot 𝑡)2
i. 1 −
1
sin 𝜃−cos 𝜃
=
1
1
+
sin 𝑡−1
sin 𝑡+1
sin2 𝑥
1+cos 𝑥
= −2 tan 𝑡 sec 𝑡
= cos 𝑥
2 sin 𝜃
sin4 𝜃−cos4 𝜃
7.5 – Sum and difference formulas:
9.
Use a sum or difference formula to find the exact value of the expression.
a. cos(45∘ + 30∘ )
b. sin 195∘
f. sin 80∘ cos 50∘ − sin 50∘ cos 80∘
4𝜋
3
c. tan �
𝜋
4
− �
d. tan
5𝜋
12
e. cos 65∘ cos 5∘ + sin 65∘ sin 5∘
10.
Suppose tan 𝛼 = −3 with 𝛼 in quadrant II, and cot 𝛽 = −3 with 𝛽 in quadrant IV, find the following.
a. sin(𝛼 + 𝛽)
11.
12.
b. cos(𝛼 − 𝛽)
1
3
Suppose sin 𝛼 = − with 𝜋 < 𝛼 <
a. sin(𝛼 + 𝛽)
Verify each identity.
𝜋
6
3𝜋
,
2
b. cos(𝛼 − 𝛽)
𝜋
3
c. tan(𝛼 + 𝛽)
c. tan(𝛼 + 𝛽)
a. sin �𝑥 + � − cos �𝑥 + � = √3 sin 𝑥
d.
cos(𝛼−𝛽)
cos 𝛼 cos 𝛽
= 1 + tan 𝛼 tan 𝛽
1
3
and cos 𝛽 = − with 15𝜋 < 4𝛽 + 3𝜋 < 17𝜋, find the following.
b. tan �𝑥 +
3𝜋
�
4
=
tan 𝑥−1
1+tan 𝑥
c. sec(𝛼 + 𝛽) =
sec 𝛼 sec 𝛽
1−tan 𝛼 tan 𝛽