Trigonometry:
Perencevich
Chap. 14 Mid-chapter Review
Gonzaga College High School
1. Evaluate the six trigonometric functions
on the angle θ illustrated below. Give
exact- value answers.
7. Find all solutions of the following
trigonometric equations in the interval 0 ≤ θ < 2π. Where possible, give exact-value answers. For
these equations it will be helpful
to recall cos2 θ + sin2 θ = 1
a) cos2 θ + cos θ = sin2 θ
b) sin2 θ = cos2 θ
c) sin2 θ + 2 cos θ + 1 = 0
d) cos2 θ = −4 sin θ
e) 6 cos2 θ − 3.1 cos θ − 2.3 = 0
2. Find the radian measure of the angles
whose degree measure is given below.
a) 120◦
b) 45◦
c) 210◦
c) 315◦
3. Give the exact value of the following:
a) cos 120◦
b) sin 30◦
c) sin 45◦
d) tan 210◦
e) sin 315◦
f) tan 180◦
4. Evaluate without a calculator:
√
a) cos−1 1/2
b) sin−1 3/2
c) sin−1 (−1/2)
√
√
d) tan−1 (− 3) e) sin−1 (−1/ 2) f) tan−1 (−1)
5. Find all solutions of the following trigonometric equations in the interval 0 ≤ θ < 2π.
Where possible, give exact-value answers.
a) 5 sin θ + 3 = 0
c) 3 sin2 θ − sin θ − 2 = 0
e) tan3 θ = tan θ
b) 4 sin2 θ − 2 = 0
d) sin2 θ−2 sin θ−1 = 0
f) 2 sin θ cos θ = cos θ
8. Find four angles θ that have the
following angles for a reference angle with 0 ≤ θ < 2π.
a) π/5
b) 0.1341
9. Sketch one period of each of the
following equations. Give the period, phase angle, and, where appropriate, the amplitude.
a) y = 3 sin(2t + π)
b) x = 4 cos(3t + π)
c) z = tan 3t
10. Give the exact value of the following:
17π
3
25π
tan
3
13π
4
19π
sin
2
a) cos
b) sin
c) cos 22π
d)
e)
f) cos
6. If sin θ = 2/3 and π/2 < θ < π, find:
a) cos θ
b) tan θ
c
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17π
6
Trigonometry:
1.
√
sin θ = 2 2/3
√
csc θ = 3/2 2
cos θ = 1/3
√
tan θ = 2 2
sec θ = 3
√
cot θ = 1/2 2
2.
a) 2π/3
c) 7π/6
b) π/4
c) 7π/4
3.
a) −1/2
b) 1/2
d) 1/ 3
e) − 2/2
√
4.
a) π/3
d) π/3
Perencevich
Chap. 14 Mid-chapter Review
√
b) π/3
e) −π/4
c)
√
2/2
f) 0
c) −π/6
f) −π/4
5. Find all solutions of the following trigonometric equations in the interval 0 ≤ θ < 2π.
Where possible, give exact-value answers.
a) θ = 3.7851
b) θ = π/4; θ = 3π/4
θ = 5π/4; θ = 7π/4
θ = 5.6397
c) θ = π/2
d) θ = 3.5687
θ = 3.8713; θ = 5.5535
e) θ = 0; θ = π
θ = π/4; θ = 3π/4
θ = 5π/4; θ = 7π/4
θ = 5.8561
f) θ = π/2; θ = 3π/2
θ = π/6; θ = 5π/6
6. If sin θ = 2/3 and π/2 < θ < π, find:
√
a) cos θ = − 5/3
√
7. Find all solutions of the following
trigonometric equations in the interval 0 ≤ θ < 2π. Where possible, give exact-value answers. For
these equations it will be helpful
to recall cos2 θ + sin2 θ = 1
a) θ = π/3; θ = π; θ = 5π/3
b) θ = π/4; θ = 3π/4
θ = 5π/4; θ = 7π/4
c) θ = 2.3921
d) θ = 3.3799
θ = 5.5337
θ = 6.0449
e) θ = 0.3785; θ = 1.9960
θ = 4.2872; θ = 5.9047
8.
a) θ = π/5; θ = 4π/5
θ = 6π/5; θ = 9π/5
b) θ = 0.1341; θ = 3.0075
θ = 3.2757; θ = 6.1491
9.
a)
b)
c)
See below for the sketches.
A = 4; period= π ; ϕ = −π/2
A = 3; period= 2π/3; ϕ = −π/3
period= π/3; ϕ = 0
10. Give the exact value of the following:
a)
d)
1
2
√
3
b) tan θ = −2/ 5
9.
c
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b)
√
− 2
2
e) −1
c) 1
f)
√
− 3
2