Trigonometry Study Resources for Competitive Exams

Trigonometry

Graphs of Trigonometric Functions Heights and Distances Applications Trigonometric Identities and Equations Trigonometric Ratios

Q. A 12-meter tall building casts a shadow of 8 meters. What is the angle of elevation of the sun?

A. 36.87 degrees
B. 45 degrees
C. 53.13 degrees
D. 30 degrees

Solution

Using tan(θ) = height/shadow = 12/8 = 1.5, θ = tan⁻¹(1.5) ≈ 36.87 degrees.

Correct Answer: A — 36.87 degrees

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Q. A building is 20 meters tall. If the angle of elevation from a point 10 meters away from the base of the building is θ, what is tan(θ)?

A. 2
B. 0.5
C. 1
D. 1.5

Solution

tan(θ) = height/distance = 20/10 = 2.

Correct Answer: A — 2

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Q. A building is 20 meters tall. If the angle of elevation from a point on the ground 10 meters away from the base of the building is θ, what is tan(θ)?

A. 2
B. 0.5
C. 1
D. 1.5

Solution

tan(θ) = opposite/adjacent = 20/10 = 2.

Correct Answer: A — 2

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Q. A kite is flying at a height of 40 meters. 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?

A. 20√3 meters
B. 40 meters
C. 30 meters
D. 50 meters

Solution

Using tan(60) = 40/distance, distance = 40/tan(60) = 40/√3 = 20√3 meters.

Correct Answer: A — 20√3 meters

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Q. A ladder 10 meters long leans against a wall. If the foot of the ladder is 6 meters away from the wall, what is the height at which the ladder touches the wall?

A. 8 meters
B. 6 meters
C. 10 meters
D. 4 meters

Solution

Using Pythagorean theorem: height = √(10² - 6²) = √(100 - 36) = √64 = 8 meters.

Correct Answer: A — 8 meters

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Q. A ladder 25 meters long leans against a wall. If the foot of the ladder is 7 meters from the wall, what is the angle of elevation of the ladder?

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

Solution

Using cos(θ) = adjacent/hypotenuse, cos(θ) = 7/25. Therefore, θ = cos⁻¹(7/25) which is approximately 60 degrees.

Correct Answer: C — 60 degrees

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Q. A person is standing 12 meters away from a vertical pole. If the angle of elevation to the top of the pole is 45 degrees, what is the height of the pole?

A. 12 meters
B. 6 meters
C. 8 meters
D. 10 meters

Solution

Using tan(45) = height / 12, height = 12 * tan(45) = 12 * 1 = 12 meters.

Correct Answer: A — 12 meters

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Q. A person is standing 25 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?

A. 25√3 meters
B. 15 meters
C. 20 meters
D. 30 meters

Solution

Using tan(60°) = height/25, height = 25 * √3 = 25√3 meters.

Correct Answer: A — 25√3 meters

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Q. A person standing 30 meters away from a building observes the top of the building at an angle of elevation of 60 degrees. How tall is the building?

A. 15√3 meters
B. 30 meters
C. 20 meters
D. 25 meters

Solution

Using tan(60°) = height/30, height = 30 * √3 = 30√3 meters.

Correct Answer: A — 15√3 meters

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Q. A person standing 50 meters away from a building observes the top of the building at an angle of elevation of 60 degrees. How tall is the building?

A. 25√3 meters
B. 50 meters
C. 30 meters
D. 40 meters

Solution

Using tan(60) = height / 50, height = 50 * tan(60) = 50 * √3 = 25√3 meters.

Correct Answer: A — 25√3 meters

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Q. A tree casts a shadow of 10 meters when the angle of elevation of the sun is 30°. How tall is the tree?

A. 5 m
B. 10 m
C. 15 m
D. 20 m

Solution

Using tan(30°) = height/shadow, height = 10 * tan(30°) = 10 * (1/√3) = 10 m.

Correct Answer: B — 10 m

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Q. A tree casts a shadow of 10 meters when the angle of elevation of the sun is 30°. What is the height of the tree?

A. 5√3
B. 10
C. 10√3
D. 15

Solution

Using tan(30°) = height/shadow, height = 10 * tan(30°) = 10 * (1/√3) = 10√3/3.

Correct Answer: A — 5√3

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Q. A tree casts a shadow of 15 meters when the angle of elevation of the sun is 30 degrees. How tall is the tree?

A. 5√3 meters
B. 15 meters
C. 10 meters
D. 7.5 meters

Solution

Using the tangent function, tan(30) = height / 15. Therefore, height = 15 * tan(30) = 15 * (1/√3) = 5√3 meters.

Correct Answer: A — 5√3 meters

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Q. At what value of x does the function y = tan(x) have a vertical asymptote?

A. 0
B. π/4
C. π/2
D. π

Solution

The tangent function has vertical asymptotes at x = π/2 + nπ, where n is an integer. The first asymptote in [0, π] is at π/2.

Correct Answer: C — π/2

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Q. For the function y = sin(3x), what is the period?

A. π
B. 2π
C. 3π
D. 6π

Solution

The period is calculated as 2π divided by the coefficient of x, which is 3, giving a period of 2π/3.

Correct Answer: A — π

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Q. From a point 25 meters away from the base of a building, the angle of elevation to the top of the building is 30 degrees. What is the height of the building?

A. 25/√3 meters
B. 15 meters
C. 20 meters
D. 10 meters

Solution

Using tan(30) = height / 25, height = 25 * tan(30) = 25 * (1/√3) = 25/√3 meters.

Correct Answer: A — 25/√3 meters

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Q. From a point 30 meters away from the base of a tower, the angle of elevation to the top of the tower is 45 degrees. What is the height of the tower?

A. 30 meters
B. 45 meters
C. 60 meters
D. 15 meters

Solution

Using tan(45) = height / 30, we have height = 30 * tan(45) = 30 * 1 = 30 meters.

Correct Answer: A — 30 meters

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Q. From a point on the ground, the angle of elevation to the top of a 15-meter tall building is 45 degrees. How far is the point from the base of the building?

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

Solution

Using tan(45°) = height/distance, distance = height/tan(45°) = 15/1 = 15 meters.

Correct Answer: A — 15 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 point is 50 meters away from the base of the hill, what is the height of the hill?

A. 50 meters
B. 25 meters
C. 70 meters
D. 45 meters

Solution

Using tan(45°) = height/50, height = 50 * tan(45°) = 50 meters.

Correct Answer: A — 50 meters

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Q. If a tree casts a shadow of 10 meters and the angle of elevation to the top of the tree is 45 degrees, what is the height of the tree?

A. 5
B. 10
C. 15
D. 20

Solution

height = shadow * tan(45°) = 10 * 1 = 10 meters

Correct Answer: B — 10

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Q. If cos(θ) = 0.6, what is sin(θ) using the Pythagorean identity?

A. 0.8
B. 0.6
C. 0.4
D. 0.2

Solution

Using sin²θ + cos²θ = 1, sin²θ = 1 - 0.6² = 0.64, so sin(θ) = 0.8.

Correct Answer: A — 0.8

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Q. If cos(θ) = 0.8, what is the value of θ in degrees (to the nearest degree)?

A. 36
B. 45
C. 60
D. 64

Solution

θ = arccos(0.8) ≈ 36.87°, rounded to 37°

Correct Answer: D — 64

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Q. If sin(x) = 0.5, what are the possible values of x in degrees?

A. 30°, 150°
B. 45°, 135°
C. 60°, 120°
D. 90°, 270°

Solution

sin(x) = 0.5 gives x = 30° and 150°.

Correct Answer: A — 30°, 150°

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Q. If sin(x) = 0.8, what is the value of csc(x)?

A. 0.8
B. 1.25
C. 1.2
D. 0.5

Solution

csc(x) = 1/sin(x) = 1/0.8 = 1.25.

Correct Answer: B — 1.25

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Q. If sin(θ) = 0.6, what is cos(θ) using the Pythagorean identity?

A. 0.8
B. 0.6
C. 0.4
D. 0.2

Solution

Using sin²θ + cos²θ = 1, cos²θ = 1 - 0.6² = 0.64, thus cos(θ) = 0.8.

Correct Answer: A — 0.8

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Q. If sin(θ) = 0.6, what is the value of cos(θ) using the Pythagorean identity?

A. 0.8
B. 0.6
C. 0.4
D. 0.2

Solution

cos(θ) = √(1 - sin²(θ)) = √(1 - 0.36) = √0.64 = 0.8

Correct Answer: A — 0.8

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Q. If sin(θ) = 0.6, what is θ in degrees (to the nearest degree)?

A. 36
B. 45
C. 60
D. 70

Solution

θ = arcsin(0.6) ≈ 36.87°, rounded to 37°

Correct Answer: D — 70

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Q. If sin(θ) = 0.8, what is the value of θ in degrees?

A. 30
B. 45
C. 53.13
D. 60

Solution

θ = sin⁻¹(0.8) ≈ 53.13°

Correct Answer: C — 53.13

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Q. If tan(x) = 1, what is the value of x in the interval [0°, 360°)?

A. 45°
B. 135°
C. 225°
D. 315°

Solution

tan(x) = 1 at x = 45° and 225°; in the interval [0°, 360°), the first solution is 45°.

Correct Answer: A — 45°

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Q. If tan(θ) = 3/4, what is sin(θ)?

A. 3/5
B. 4/5
C. 5/5
D. 1

Solution

Using the identity tan(θ) = sin(θ)/cos(θ), we find sin(θ) = 3/5.

Correct Answer: A — 3/5

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Trigonometry

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