Q. Solve the equation sin(3x) = 0 for x in the interval [0, 2π].
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A.
0, π, 2π
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B.
0, π/3, 2π/3
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C.
0, π/2, π
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D.
0, π/4, π/2
Solution
The solutions are x = 0, π, 2π, and x = nπ/3 for n = 0, 1, 2, 3, 4, 5.
Correct Answer: A — 0, π, 2π
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Q. Solve the equation sin(x) = 0.5 for x in the interval [0, 2π].
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A.
π/6
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B.
5π/6
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C.
7π/6
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D.
11π/6
Solution
The solutions are x = π/6 and x = 5π/6 in the interval [0, 2π].
Correct Answer: A — π/6
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Q. Solve the equation tan(x) = √3 for x in the interval [0, 2π].
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A.
π/3
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B.
2π/3
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C.
4π/3
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D.
5π/3
Solution
The solutions are x = π/3 and x = 4π/3 in the interval [0, 2π].
Correct Answer: A — π/3
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Q. Solve the equation tan^2(x) = 3 for x in the interval [0, 2π].
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A.
π/3
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B.
2π/3
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C.
4π/3
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D.
5π/3
Solution
The solutions are x = π/3 and x = 4π/3.
Correct Answer: A — π/3
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Q. Solve the equation tan^2(x) = 3.
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A.
x = π/3
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B.
x = 2π/3
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C.
x = 4π/3
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D.
x = 5π/3
Solution
The solutions are x = π/3 and x = 4π/3.
Correct Answer: A — x = π/3
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Q. What are the solutions of the equation cos(x) + sin(x) = 1?
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A.
x = 0
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B.
x = π/4
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C.
x = π/2
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D.
x = π
Solution
The only solution is x = 0.
Correct Answer: A — x = 0
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Q. What are the solutions of the equation cos(x) = -1/2 in the interval [0, 2π]?
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A.
2π/3, 4π/3
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B.
π/3, 5π/3
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C.
π/2, 3π/2
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D.
0, π
Solution
The solutions are x = 2π/3 and 4π/3.
Correct Answer: A — 2π/3, 4π/3
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Q. What are the solutions of the equation cos(x) = -1/2?
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A.
2π/3
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B.
4π/3
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C.
π/3
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D.
5π/3
Solution
The solutions are x = 2π/3 and x = 4π/3.
Correct Answer: A — 2π/3
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Q. What are the solutions of the equation cos^2(x) - 1/2 = 0?
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A.
x = π/4
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B.
x = 3π/4
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C.
x = 5π/4
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D.
x = 7π/4
Solution
The solutions are x = π/4, 3π/4, 5π/4, and 7π/4.
Correct Answer: A — x = π/4
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Q. What are the solutions of the equation sin(2x) = 0 in the interval [0, 2π]?
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A.
0, π, 2π
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B.
0, π/2, π
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C.
0, π/4, π/2
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D.
0, 3π/2
Solution
The solutions are x = 0, π, and 2π.
Correct Answer: A — 0, π, 2π
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Q. What are the solutions of the equation sin(2x) = 0?
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A.
x = nπ/2
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B.
x = nπ
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C.
x = nπ + π/2
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D.
x = nπ + π
Solution
sin(2x) = 0 implies 2x = nπ, thus x = nπ/2 for n ∈ Z.
Correct Answer: A — x = nπ/2
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Q. What are the solutions of the equation sin(x) = sin(π/3)?
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A.
x = π/3
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B.
x = 2π/3
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C.
x = 4π/3
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D.
x = 5π/3
Solution
The solutions are x = π/3 + 2nπ and x = 2π/3 + 2nπ.
Correct Answer: A — x = π/3
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Q. What are the solutions of the equation sin^2(x) - sin(x) = 0?
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A.
x = nπ
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B.
x = nπ + π/2
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C.
x = nπ + 2π
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D.
x = nπ + π
Solution
Factoring gives sin(x)(sin(x) - 1) = 0, so sin(x) = 0 or sin(x) = 1. Thus, x = nπ or x = π/2 + 2nπ.
Correct Answer: A — x = nπ
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