Q. Determine the values of x that satisfy cos^2(x) - 1/2 = 0.
-
A.
π/4, 3π/4
-
B.
π/3, 2π/3
-
C.
π/6, 5π/6
-
D.
0, π
Solution
The solutions are x = π/4 and x = 3π/4.
Correct Answer: A — π/4, 3π/4
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Q. Determine the values of x that satisfy sin^2(x) - sin(x) = 0.
-
A.
0, π
-
B.
0, π/2, π
-
C.
0, π/2, 3π/2
-
D.
0, π/2, π, 3π/2
Solution
The solutions are x = 0, π/2, π, 3π/2.
Correct Answer: D — 0, π/2, π, 3π/2
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Q. Determine the values of x that satisfy the equation sin(2x) = 0.
-
A.
x = nπ/2
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B.
x = nπ
-
C.
x = nπ/4
-
D.
x = nπ/3
Solution
The solutions are x = nπ/2, where n is any integer.
Correct Answer: A — x = nπ/2
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Q. Determine the values of x that satisfy the equation sin^2(x) - sin(x) = 0.
-
A.
0, π
-
B.
0, π/2, π
-
C.
0, π/2, 3π/2
-
D.
0, π/2, π, 3π/2
Solution
Factoring gives sin(x)(sin(x) - 1) = 0, so x = 0, π/2, π, 3π/2.
Correct Answer: D — 0, π/2, π, 3π/2
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Q. Find the general solution of the equation cos(2x) = 0.
-
A.
x = (2n+1)π/4
-
B.
x = nπ/2
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C.
x = (2n+1)π/2
-
D.
x = nπ
Solution
The general solution is x = (2n+1)π/4, where n is any integer.
Correct Answer: A — x = (2n+1)π/4
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Q. Find the general solution of the equation sin(x) + sin(2x) = 0.
-
A.
x = nπ
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B.
x = nπ/2
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C.
x = (2n+1)π/4
-
D.
x = nπ/3
Solution
Factoring gives sin(x)(1 + 2cos(x)) = 0, leading to x = nπ or cos(x) = -1/2.
Correct Answer: A — x = nπ
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Q. Find the general solution of the equation sin(x) + √3 cos(x) = 0.
-
A.
x = (2n+1)π/3
-
B.
x = (2n+1)π/6
-
C.
x = nπ
-
D.
x = (2n+1)π/4
Solution
The general solution is x = (2n+1)π/3, where n is an integer.
Correct Answer: A — x = (2n+1)π/3
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Q. Find the general solution of the equation sin(x) + √3cos(x) = 0.
-
A.
x = (2n+1)π/3
-
B.
x = nπ
-
C.
x = (2n+1)π/4
-
D.
x = nπ + π/6
Solution
The general solution is x = (2n+1)π/3, where n is an integer.
Correct Answer: A — x = (2n+1)π/3
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Q. Find the general solution of the equation sin(x) = -1/2.
-
A.
x = 7π/6 + 2nπ
-
B.
x = 11π/6 + 2nπ
-
C.
x = 7π/6, 11π/6
-
D.
Both 1 and 2
Solution
The general solutions are x = 7π/6 + 2nπ and x = 11π/6 + 2nπ.
Correct Answer: D — Both 1 and 2
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Q. Find the general solution of the equation sin(x) = sin(2x).
-
A.
x = nπ
-
B.
x = nπ/3
-
C.
x = nπ/2
-
D.
x = nπ/4
Solution
Using the identity sin(a) = sin(b) gives x = nπ or x = (2n+1)π/3.
Correct Answer: A — x = nπ
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Q. Find the general solution of the equation sin(x) = sin(π/4).
-
A.
x = nπ + (-1)^n π/4
-
B.
x = nπ + π/4
-
C.
x = nπ + 3π/4
-
D.
x = nπ + π/2
Solution
The general solution is x = nπ + (-1)^n π/4, where n is any integer.
Correct Answer: A — x = nπ + (-1)^n π/4
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Q. Find the solutions of the equation 2sin(x) + √3 = 0.
-
A.
x = 5π/6
-
B.
x = 7π/6
-
C.
x = π/6
-
D.
x = 11π/6
Solution
Solving gives sin(x) = -√3/2, so x = 7π/6 and 11π/6.
Correct Answer: B — x = 7π/6
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Q. Find the solutions of the equation 2sin(x) - 1 = 0 in the interval [0, 2π].
-
A.
π/6, 5π/6
-
B.
π/4, 3π/4
-
C.
π/3, 2π/3
-
D.
π/2, 3π/2
Solution
The solutions are x = π/6 and x = 5π/6.
Correct Answer: A — π/6, 5π/6
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Q. Find the solutions of the equation 2sin(x) - 1 = 0.
-
A.
π/6
-
B.
5π/6
-
C.
7π/6
-
D.
11π/6
Solution
The solutions are x = π/6 and x = 5π/6.
Correct Answer: A — π/6
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Q. Find the values of x that satisfy 3cos^2(x) - 1 = 0.
-
A.
π/3, 2π/3
-
B.
0, π
-
C.
π/2, 3π/2
-
D.
0, 2π
Solution
Solving gives cos^2(x) = 1/3, so x = π/3, 2π/3, and their equivalents.
Correct Answer: A — π/3, 2π/3
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Q. Find the values of x that satisfy 3sin(x) - 1 = 0.
-
A.
π/6
-
B.
5π/6
-
C.
7π/6
-
D.
11π/6
Solution
The solution is x = π/6 + 2nπ.
Correct Answer: A — π/6
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Q. Find the values of x that satisfy sin^2(x) - sin(x) - 2 = 0.
-
A.
-1, 2
-
B.
1, -2
-
C.
2, -1
-
D.
0, 1
Solution
Factoring gives (sin(x) - 2)(sin(x) + 1) = 0, so sin(x) = 2 (not possible) or sin(x) = -1, giving x = 3π/2.
Correct Answer: A — -1, 2
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Q. Find the values of x that satisfy sin^2(x) - sin(x) = 0.
-
A.
0, π
-
B.
0, π/2
-
C.
0, 2π
-
D.
0, 3π/2
Solution
Factoring gives sin(x)(sin(x) - 1) = 0, so x = 0 and x = π.
Correct Answer: A — 0, π
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Q. Find the values of x that satisfy the equation 3sin(x) - 1 = 0.
-
A.
π/6
-
B.
5π/6
-
C.
7π/6
-
D.
11π/6
Solution
Rearranging gives sin(x) = 1/3, which has solutions in the specified interval.
Correct Answer: A — π/6
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Q. Find the values of x that satisfy the equation 3sin(x) - 2 = 0.
-
A.
π/6
-
B.
5π/6
-
C.
π/2
-
D.
7π/6
Solution
The solution is x = π/2.
Correct Answer: C — π/2
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Q. Find the values of x that satisfy the equation sin^2(x) - sin(x) = 0.
-
A.
0, π
-
B.
0, π/2
-
C.
0, 2π
-
D.
0, 3π/2
Solution
Factoring gives sin(x)(sin(x) - 1) = 0, so x = 0 and x = π.
Correct Answer: A — 0, π
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Q. Solve the equation 2sin(x) + √3 = 0 for x in the interval [0, 2π].
-
A.
5π/3
-
B.
π/3
-
C.
2π/3
-
D.
4π/3
Solution
Rearranging gives sin(x) = -√3/2, so x = 4π/3 and x = 5π/3.
Correct Answer: A — 5π/3
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Q. Solve the equation 2sin(x) - 1 = 0 for x in the interval [0, 2π].
-
A.
π/6
-
B.
5π/6
-
C.
π/2
-
D.
7π/6
Solution
The solution is x = π/2.
Correct Answer: C — π/2
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Q. Solve the equation 3cos^2(x) - 1 = 0.
-
A.
x = π/3, 2π/3
-
B.
x = π/4, 3π/4
-
C.
x = 0, π
-
D.
x = π/6, 5π/6
Solution
Rearranging gives cos^2(x) = 1/3, so x = π/3 and 2π/3.
Correct Answer: A — x = π/3, 2π/3
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Q. Solve the equation 3sin(x) - 4 = 0 for x in the interval [0, 2π].
-
A.
π/6
-
B.
π/3
-
C.
2π/3
-
D.
5π/6
Solution
The solution is x = π/3.
Correct Answer: B — π/3
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Q. Solve the equation cos(x) + sin(x) = 1 for x in the interval [0, 2π].
-
A.
π/4
-
B.
π/2
-
C.
3π/4
-
D.
0
Solution
The only solution is x = π/2.
Correct Answer: B — π/2
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Q. Solve the equation cos(x) = -1/2 for x in the interval [0, 2π].
-
A.
2π/3, 4π/3
-
B.
π/3, 5π/3
-
C.
π/2, 3π/2
-
D.
0, π
Solution
The solutions are x = 2π/3 and x = 4π/3 in the interval [0, 2π].
Correct Answer: A — 2π/3, 4π/3
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Q. Solve the equation sin(2x) = 0 for x in the interval [0, 2π].
-
A.
0, π, 2π
-
B.
π/2, 3π/2
-
C.
π/4, 3π/4
-
D.
π/6, 5π/6
Solution
The solutions are x = 0, π, 2π.
Correct Answer: A — 0, π, 2π
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Q. Solve the equation sin(2x) = 1 for x in the interval [0, 2π].
-
A.
π/4
-
B.
3π/4
-
C.
π/2
-
D.
5π/4
Solution
The equation sin(2x) = 1 gives 2x = π/2 + 2nπ, hence x = π/4 + nπ/2. In [0, 2π], the solutions are π/4 and 5π/4.
Correct Answer: C — π/2
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Q. Solve the equation sin(2x) = √3/2 for x in the interval [0, 2π].
-
A.
π/12
-
B.
5π/12
-
C.
7π/12
-
D.
11π/12
Solution
The solutions are x = π/12, 5π/12, 7π/12, and 11π/12.
Correct Answer: A — π/12
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