Trigonometry
Q. A tower is 50 meters high. From a point on the ground, the angle of elevation to the top of the tower is 30 degrees. What is the distance from the point to the base of the tower?
A.
25√3 m
B.
50 m
C.
25 m
D.
50√3 m
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Solution
Using tan(30°) = height/distance, we have distance = height/tan(30°) = 50/√3 = 25√3 m.
Correct Answer: A — 25√3 m
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Q. A tower is 60 meters high. From a point on the ground, the angle of elevation to the top of the tower is 45 degrees. How far is the point from the base of the tower?
A.
30 meters
B.
60 meters
C.
45 meters
D.
75 meters
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Solution
Distance = height / tan(angle) = 60 / 1 = 60 meters.
Correct Answer: B — 60 meters
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Q. A tower is 80 meters high. From a point on the ground, the angle of elevation to the top of the tower is 60 degrees. How far is the point from the base of the tower?
A.
40 m
B.
80 m
C.
20 m
D.
60 m
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Solution
Distance = height / tan(angle) = 80 / √3 = 40 m.
Correct Answer: A — 40 m
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Q. A tower is standing on a horizontal ground. The angle of elevation of the top of the tower from a point on the ground is 30 degrees. If the height of the tower is 10√3 m, how far is the point from the base of the tower?
A.
10 m
B.
5 m
C.
15 m
D.
20 m
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Solution
Using tan(30°) = height/distance, we have 1/√3 = 10√3/distance. Therefore, distance = 10√3 * √3 = 30 m.
Correct Answer: A — 10 m
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Q. A tower is standing on a horizontal ground. The angle of elevation of the top of the tower from a point on the ground is 30 degrees. If the height of the tower is 10√3 meters, how far is the point from the base of the tower?
A.
10 m
B.
20 m
C.
30 m
D.
40 m
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Solution
Using tan(30°) = height/distance, we have 1/√3 = 10√3/distance. Therefore, distance = 10√3 * √3 = 30 m.
Correct Answer: B — 20 m
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Q. A tower is standing on a horizontal ground. The angle of elevation of the top of the tower from a point on the ground is 30 degrees. If the height of the tower is 50 meters, how far is the point from the base of the tower?
A.
50√3 m
B.
100 m
C.
50 m
D.
100√3 m
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Solution
Using tan(30°) = height/distance, we have distance = height/tan(30°) = 50/(1/√3) = 50√3 m.
Correct Answer: A — 50√3 m
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Q. A tree casts a shadow of 20 m when the angle of elevation of the sun is 30 degrees. What is the height of the tree?
A.
10 m
B.
15 m
C.
20 m
D.
25 m
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Solution
Using tan(30°) = height/20, we have 1/√3 = height/20. Therefore, height = 20/√3 ≈ 11.55 m.
Correct Answer: A — 10 m
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Q. A tree casts a shadow of 20 m when the angle of elevation of the sun is 45 degrees. What is the height of the tree?
A.
10 m
B.
20 m
C.
30 m
D.
40 m
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Solution
Using tan(45°) = height/shadow, we have 1 = height/20. Therefore, height = 20 m.
Correct Answer: B — 20 m
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Q. A tree casts a shadow of 20 meters when the angle of elevation of the sun is 30 degrees. What is the height of the tree?
A.
20√3 meters
B.
10√3 meters
C.
30 meters
D.
40 meters
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Solution
Height = shadow * tan(angle) = 20 * tan(30°) = 20 * (1/√3) = 20√3 meters.
Correct Answer: A — 20√3 meters
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Q. A tree is 15 meters tall. From a point on the ground, the angle of elevation to the top of the tree is 30 degrees. How far is the point from the base of the tree?
A.
15√3 meters
B.
30 meters
C.
45 meters
D.
10 meters
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Solution
Distance = height / tan(angle) = 15 / tan(30°) = 15√3 meters.
Correct Answer: A — 15√3 meters
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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
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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
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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
B.
x = nπ
C.
x = nπ/4
D.
x = nπ/3
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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
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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
C.
x = (2n+1)π/2
D.
x = nπ
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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
B.
x = nπ/2
C.
x = (2n+1)π/4
D.
x = nπ/3
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Solution
The solution is x = π/2.
Correct Answer: C — π/2
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