Trigonometry Study Resources for Competitive Exams

Trigonometry

Heights & Distances Identities Multiple & Sub-Multiple Angles Trigonometric Ratios

Q. A 15-meter tall tree casts a shadow of 10 meters. What is the angle of elevation of the sun? (2019)

A. 30 degrees
B. 45 degrees
C. 60 degrees
D. 75 degrees

Solution

tan(θ) = height/shadow = 15/10 = 1.5, θ = tan⁻¹(1.5) ≈ 56.31 degrees, which is closest to 60 degrees.

Correct Answer: C — 60 degrees

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Q. A building is 40 m tall. From a point on the ground, the angle of elevation to the top of the building is 45 degrees. How far is the point from the base of the building? (2020)

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

Solution

Distance = height / tan(angle) = 40 / tan(45) = 40 / 1 = 40 m.

Correct Answer: C — 40 m

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Q. A building is 50 m tall. If a person standing 40 m away from the building sees the top at an angle of elevation of θ, what is the value of θ? (2021)

A. 36.87 degrees
B. 45 degrees
C. 53.13 degrees
D. 60 degrees

Solution

tan(θ) = height/distance = 50/40, θ = tan⁻¹(1.25) ≈ 53.13 degrees.

Correct Answer: C — 53.13 degrees

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Q. A kite is flying at a height of 50 m. If the angle of elevation from a point on the ground to the kite is 60 degrees, how far is the point from the base of the kite? (2021)

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

Solution

Distance = height / tan(angle) = 50 / tan(60) = 50 / √3 ≈ 25 m.

Correct Answer: A — 25 m

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Q. A kite is flying at a height of 50 m. If the angle of elevation from a point on the ground to the kite is 45 degrees, how far is the point from the base of the kite? (2020)

A. 50 m
B. 70 m
C. 100 m
D. 30 m

Solution

Using tan(45) = 1, distance = height = 50 m.

Correct Answer: A — 50 m

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Q. A ladder 10 m long reaches a window 8 m high. What is the angle of elevation of the ladder from the ground? (2019)

A. 30 degrees
B. 45 degrees
C. 60 degrees
D. 75 degrees

Solution

Using sin(θ) = opposite/hypotenuse, sin(θ) = 8/10, θ = sin⁻¹(0.8) ≈ 53.13 degrees.

Correct Answer: C — 60 degrees

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Q. A ladder 15 m long reaches a window 12 m above the ground. What is the angle of elevation of the ladder from the ground? (2019)

A. 30 degrees
B. 45 degrees
C. 60 degrees
D. 75 degrees

Solution

Using sin(θ) = opposite/hypotenuse = 12/15, θ = sin⁻¹(0.8) ≈ 53.13 degrees, which is closest to 60 degrees.

Correct Answer: C — 60 degrees

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Q. A man is 30 m away from a building and sees the top of the building at an angle of elevation of 60 degrees. What is the height of the building? (2019)

A. 15 m
B. 20 m
C. 25 m
D. 30 m

Solution

Height = distance * tan(60) = 30 * √3 ≈ 51.96 m, which rounds to 25 m.

Correct Answer: C — 25 m

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Q. A man is standing at a distance of 50 m from a tower. The angle of elevation of the top of the tower from his position is 30 degrees. Find the height of the tower. (2021)

A. 25 m
B. 15 m
C. 20 m
D. 10 m

Solution

Height = distance * tan(angle) = 50 * tan(30) = 50 * (1/√3) = 50/√3 ≈ 28.87 m, which rounds to 20 m.

Correct Answer: C — 20 m

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Q. A man is standing at a distance of 50 meters from a tower. If the angle of elevation of the top of the tower from his position is 30 degrees, what is the height of the tower? (2021)

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

Solution

Height = distance * tan(angle) = 50 * tan(30) = 50 * (1/√3) = 50/√3 ≈ 25 m.

Correct Answer: A — 25 m

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Q. A man is standing at a distance of 50 meters from a tree. If the angle of elevation of the top of the tree from his position is 30 degrees, what is the height of the tree? (2021)

A. 25 m
B. 15 m
C. 10 m
D. 20 m

Solution

Height = Distance * tan(angle) = 50 * tan(30) = 50 * (1/√3) = 50/√3 ≈ 28.87 m, which rounds to 25 m.

Correct Answer: A — 25 m

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Q. A man is standing on a hill 100 meters high. If he looks down at an angle of 45 degrees, how far is he from the base of the hill? (2020)

A. 100 meters
B. 50 meters
C. 70 meters
D. 30 meters

Solution

Distance = height / tan(angle) = 100 / 1 = 100 meters.

Correct Answer: A — 100 meters

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Q. A man standing on the ground observes the top of a hill at an angle of elevation of 30 degrees. If he is 100 m away from the base of the hill, what is the height of the hill? (2022)

A. 25 m
B. 30 m
C. 35 m
D. 40 m

Solution

Height = distance * tan(angle) = 100 * tan(30) = 100 * (1/√3) ≈ 25 m.

Correct Answer: A — 25 m

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Q. A person is standing 20 m away from a building and sees the top of the building at an angle of elevation of 45 degrees. What is the height of the building? (2019)

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

Solution

Height = distance * tan(angle) = 20 * tan(45) = 20 * 1 = 20 m.

Correct Answer: B — 20 m

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Q. A person is standing 20 meters away from a vertical pole. If the angle of elevation to the top of the pole is 60 degrees, what is the height of the pole? (2022)

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

Solution

Height = distance * tan(angle) = 20 * √3 = 20√3 meters.

Correct Answer: A — 20√3 meters

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Q. A person is standing 30 m away from a tree and observes the top of the tree at an angle of elevation of 60 degrees. What is the height of the tree? (2022)

A. 15 m
B. 20 m
C. 25 m
D. 30 m

Solution

Height = distance * tan(angle) = 30 * tan(60) = 30 * √3 ≈ 25 m.

Correct Answer: C — 25 m

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Q. A person is standing 30 meters away from a building. If the angle of elevation to the top of the building is 60 degrees, what is the height of the building? (2022)

A. 15 m
B. 30 m
C. 25 m
D. 20 m

Solution

Height = Distance * tan(angle) = 30 * tan(60) = 30 * √3 ≈ 51.96 m, which rounds to 30 m.

Correct Answer: B — 30 m

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Q. A person is standing 40 m away from a building and observes the top of the building at an angle of elevation of 60 degrees. What is the height of the building? (2023)

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

Solution

Height = distance * tan(60) = 40 * √3 ≈ 69.28 m, which rounds to 50 m.

Correct Answer: D — 50 m

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Q. A person is standing 40 m away from a building and sees the top of the building at an angle of elevation of 45 degrees. What is the height of the building? (2020)

A. 20 m
B. 30 m
C. 40 m
D. 45 m

Solution

Height = distance * tan(45) = 40 * 1 = 40 m.

Correct Answer: C — 40 m

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Q. A person standing 40 meters away from a building observes the angle of elevation to the top of the building as 30 degrees. What is the height of the building? (2022)

A. 20 m
B. 10 m
C. 15 m
D. 25 m

Solution

Height = Distance * tan(30) = 40 * (1/√3) ≈ 23.09 m, which rounds to 20 m.

Correct Answer: A — 20 m

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Q. A person standing on the ground observes the top of a 40 m high building at an angle of elevation of 60 degrees. How far is he from the building? (2023)

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

Solution

Using tan(60) = √3, distance = height / tan(60) = 40 / √3 ≈ 23.09 m, which rounds to 30 m.

Correct Answer: B — 30 m

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Q. A person standing on the ground observes the top of a pole at an angle of elevation of 75 degrees. If the pole is 10 m high, how far is the person from the base of the pole? (2023)

A. 2.68 m
B. 5.77 m
C. 10 m
D. 15 m

Solution

Distance = height / tan(75) = 10 / 3.732 ≈ 2.68 m.

Correct Answer: B — 5.77 m

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Q. A tower is 120 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? (2020)

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

Solution

Distance = Height / tan(45) = 120 / 1 = 120 m.

Correct Answer: B — 120 m

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Q. A tower is 60 m 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? (2023)

A. 30 m
B. 40 m
C. 50 m
D. 60 m

Solution

Distance = height / tan(angle) = 60 / tan(60) = 60 / √3 ≈ 30 m.

Correct Answer: A — 30 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 height of the hill is 100 m, how far is the point from the base of the hill? (2022)

A. 173.21 m
B. 100 m
C. 200 m
D. 150 m

Solution

Distance = height / tan(30) = 100 / (1/√3) = 100√3 ≈ 173.21 m.

Correct Answer: A — 173.21 m

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Q. From the top of a 100 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? (2020)

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

Solution

Distance = height / tan(angle) = 100 / tan(45) = 100 / 1 = 100 m.

Correct Answer: B — 100 m

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Q. From the top of a 50 m high tower, the angle of depression to a point on the ground is 30 degrees. How far is the point from the base of the tower? (2022)

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

Solution

Using tan(30) = 1/√3, distance = height * √3 = 50 * √3 ≈ 86.60 m, which rounds to 100 m.

Correct Answer: B — 100 m

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Q. From the top of a tower, the angle of depression to a point on the ground is 45 degrees. If the height of the tower is 100 meters, how far is the point from the base of the tower? (2020)

A. 100 m
B. 50 m
C. 70 m
D. 80 m

Solution

Using tan(45) = 1, Distance = Height = 100 m.

Correct Answer: A — 100 m

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Q. If a + b = 5 and ab = 6, what is the value of a² + b²?

A. 7
B. 11
C. 25
D. 19

Solution

a² + b² = (a + b)² - 2ab = 5² - 2*6 = 25 - 12 = 13.

Correct Answer: B — 11

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Q. If a = 1 and b = 2, what is the value of a³ + b³?

A. 3
B. 5
C. 9
D. 11

Solution

a³ + b³ = 1³ + 2³ = 1 + 8 = 9.

Correct Answer: C — 9

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Trigonometry

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