Trigonometry Study Resources for Competitive Exams

Trigonometry

Heights & Distances Properties of Triangles Trigonometric Equations Trigonometric Ratios & Identities

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. From a point A, the angle of elevation of the top of a building is 45 degrees. If the height of the building is 20 meters, how far is point A from the base of the building?

A. 10 m
B. 20 m
C. 30 m
D. 40 m

Solution

Using tan(45°) = height/distance, we have 1 = 20/distance. Therefore, distance = 20 m.

Correct Answer: B — 20 m

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Q. From a point on the ground, the angle of elevation of the top of a hill is 30 degrees. If the height of the hill is 50 meters, how far is the point from the base of the hill?

A. 50 m
B. 75 m
C. 100 m
D. 125 m

Solution

Using tan(30°) = height/distance, we have 1/√3 = 50/distance. Therefore, distance = 50√3 ≈ 86.6 m.

Correct Answer: C — 100 m

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Q. From a point on the ground, the angle of elevation of the top of a hill is 30 degrees. If the distance from the point to the base of the hill is 100 meters, what is the height of the hill?

A. 50 m
B. 60 m
C. 70 m
D. 80 m

Solution

Using tan(30°) = height/100, we have 1/√3 = height/100. Therefore, height = 100/√3 ≈ 57.74 m.

Correct Answer: A — 50 m

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Q. From a point on the ground, the angle of elevation to the top of a 40 m high building is 45 degrees. How far is the point from the base of the building?

A. 40 m
B. 20 m
C. 30 m
D. 50 m

Solution

Using tan(45°) = height/distance, we have distance = height/tan(45°) = 40/1 = 40 m.

Correct Answer: A — 40 m

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Q. From a point on the ground, the angle of elevation to the top of a 75 m high building is 45 degrees. How far is the point from the building?

A. 75 m
B. 50 m
C. 100 m
D. 25 m

Solution

Using tan(45°) = height/distance, we have distance = height/tan(45°) = 75/1 = 75 m.

Correct Answer: A — 75 m

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Q. From a point on the ground, the angle of elevation to the top of a building is 45 degrees. If the building is 50 meters tall, how far is the point from the base of the building?

A. 50 m
B. 25 m
C. 100 m
D. 75 m

Solution

Distance = height / tan(angle) = 50 / 1 = 50 m.

Correct Answer: A — 50 m

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Q. From a point on the ground, the angle of elevation to the top of a hill is 30 degrees. If the distance from the point to the base of the hill is 100 meters, what is the height of the hill?

A. 100√3 m
B. 50 m
C. 100 m
D. 50√3 m

Solution

Using tan(30°) = height/distance, we have height = distance * tan(30°) = 100 * (1/√3) = 100/√3 = 50 m.

Correct Answer: B — 50 m

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Q. From a point on the ground, the angle of elevation to the top of a hill is 45 degrees. If the height of the hill is 20 m, how far is the point from the base of the hill?

A. 20 m
B. 10 m
C. 30 m
D. 40 m

Solution

Using tan(45°) = height/distance, we have 1 = 20/distance. Therefore, distance = 20 m.

Correct Answer: A — 20 m

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Q. From a point on the ground, the angle of elevation to the top of a hill is 45 degrees. If the height of the hill is 50 m, how far is the point from the base of the hill?

A. 25 m
B. 50 m
C. 70 m
D. 100 m

Solution

Using tan(45°) = height/distance, we have 1 = 50/distance. Therefore, distance = 50 m.

Correct Answer: B — 50 m

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Q. From a point on the ground, the angle of elevation to the top of a hill is 45 degrees. If the height of the hill is 100 m, how far is the point from the base of the hill?

A. 100 m
B. 50 m
C. 200 m
D. 150 m

Solution

Using tan(45°) = height/distance, we have distance = height/tan(45°) = 100/1 = 100 m.

Correct Answer: A — 100 m

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Q. From a point on the ground, the angle of elevation to the top of a hill is 45 degrees. If the point is 100 meters away from the base of the hill, what is the height of the hill?

A. 100 meters
B. 50 meters
C. 200 meters
D. 150 meters

Solution

Height = distance * tan(angle) = 100 * tan(45°) = 100 meters.

Correct Answer: A — 100 meters

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Q. From a point on the ground, the angle of elevation to the top of a hill is 45 degrees. If the height of the hill is 40 m, how far is the point from the base of the hill?

A. 20 m
B. 40 m
C. 60 m
D. 80 m

Solution

Using tan(45°) = height/distance, we have 1 = 40/distance. Therefore, distance = 40 m.

Correct Answer: B — 40 m

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Q. From a point on the ground, the angle of elevation to the top of a tower is 60 degrees. If the tower is 30 m high, how far is the point from the base of the tower?

A. 15 m
B. 30 m
C. 20 m
D. 10 m

Solution

Using tan(60°) = height/distance, we have √3 = 30/distance. Therefore, distance = 30/√3 m.

Correct Answer: A — 15 m

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Q. From the top of a 20-meter high building, the angle of depression to a car parked on the ground is 60 degrees. How far is the car from the base of the building?

A. 10√3 meters
B. 20 meters
C. 30 meters
D. 40 meters

Solution

Distance = height / tan(angle) = 20 / tan(60°) = 20 / √3 = 10√3 meters.

Correct Answer: A — 10√3 meters

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Q. From the top of a 50 m high building, the angle of depression to a point on the ground is 45 degrees. How far is the point from the base of the building?

A. 25 m
B. 50 m
C. 70 m
D. 100 m

Solution

Using tan(45°) = height/distance, we have 1 = 50/distance. Therefore, distance = 50 m.

Correct Answer: B — 50 m

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Q. From the top of a 50-meter high building, the angle of depression to a point on the ground is 45 degrees. How far is the point from the base of the building?

A. 50 meters
B. 100 meters
C. 25 meters
D. 70 meters

Solution

Distance = height / tan(angle) = 50 / tan(45°) = 50 meters.

Correct Answer: A — 50 meters

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Q. From the top of a 60 m high building, the angle of depression to a point on the ground is 30 degrees. How far is the point from the base of the building?

A. 60√3 m
B. 30√3 m
C. 60 m
D. 30 m

Solution

Using tan(30°) = height/distance, we have distance = height/tan(30°) = 60/√3 = 60√3 m.

Correct Answer: A — 60√3 m

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Q. From the top of a 60 m high cliff, the angle of depression to a boat in the sea is 30 degrees. How far is the boat from the base of the cliff?

A. 60√3 m
B. 30√3 m
C. 60 m
D. 30 m

Solution

Using tan(30°) = height/distance, we have distance = height/tan(30°) = 60/√3 = 60√3 m.

Correct Answer: A — 60√3 m

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Q. If a person is standing 50 meters away from a building and the angle of elevation to the top of the building is 60 degrees, what is the height of the building?

A. 25√3 meters
B. 50√3 meters
C. 75 meters
D. 100 meters

Solution

Height = distance * tan(angle) = 50 * √3 = 50√3 meters.

Correct Answer: B — 50√3 meters

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Q. If cos A = 1/2, what is the value of A in degrees?

A. 30
B. 60
C. 90
D. 120

Solution

The angle A for which cos A = 1/2 is 60°.

Correct Answer: A — 30

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Q. If cos B = 1/2, what is the value of sin B?

A. √3/2
B. 1/2
C. 0
D. √2/2

Solution

Using the identity sin^2 B + cos^2 B = 1, we have sin B = sqrt(1 - (1/2)^2) = sqrt(3/4) = √3/2.

Correct Answer: A — √3/2

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Q. If cos(x) = 0, what are the possible values of x?

A. 90° + k*180°
B. k*90°
C. k*180°
D. 0° + k*360°

Solution

cos(x) = 0 at x = 90° + k*180°, where k is any integer.

Correct Answer: A — 90° + k*180°

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Q. If cos(x) = 0, what is the value of tan(x)?

A. 0
B. 1
C. undefined
D. ∞

Solution

tan(x) = sin(x)/cos(x) is undefined when cos(x) = 0.

Correct Answer: C — undefined

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Q. If cos(x) = 0, what is the value of x?

A. π/2 + nπ
B. nπ
C. 0
D. π

Solution

cos(x) = 0 at x = π/2 + nπ, where n is any integer.

Correct Answer: A — π/2 + nπ

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Q. If cos(x) = 1/2, what is the value of sin(x)?

A. √3/2
B. 1/2
C. 0
D. √2/2

Solution

Using the identity sin^2(x) + cos^2(x) = 1, we have sin^2(x) = 1 - (1/2)^2 = 1 - 1/4 = 3/4. Therefore, sin(x) = √(3/4) = √3/2.

Correct Answer: A — √3/2

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Q. If cos(α) = 1/2, what are the possible values of α in the interval [0, 2π]?

A. π/3, 5π/3
B. π/4, 3π/4
C. 0, π
D. π/6, 11π/6

Solution

The angles where cos(α) = 1/2 in the interval [0, 2π] are α = π/3 and α = 5π/3.

Correct Answer: A — π/3, 5π/3

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Q. If cos(θ) = 1/2, what are the possible values of θ in degrees?

A. 30, 150
B. 45, 135
C. 60, 120
D. 0, 180

Solution

cos(θ) = 1/2 at θ = 30° and 150°.

Correct Answer: A — 30, 150

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Q. If cos(θ) = 1/2, what are the possible values of θ in the range [0°, 360°]?

A. 60°, 300°
B. 30°, 150°
C. 90°, 270°
D. 0°, 180°

Solution

cos(θ) = 1/2 at θ = 60° and θ = 300°.

Correct Answer: A — 60°, 300°

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Q. If cos(θ) = 1/2, what are the possible values of θ?

A. 30°, 150°
B. 45°, 135°
C. 60°, 120°
D. 0°, 180°

Solution

cos(θ) = 1/2 at θ = 30° and 150°.

Correct Answer: A — 30°, 150°

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Trigonometry

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