Question 1

A ladder leans against a wall making an angle of 60° with the ground. If the ladder is 10 m long, how far is the base of the ladder from the wall?

Accepted Answer

The distance from the wall is the base of the triangle; using cos 60° = base/hypotenuse: base = 10 × (1/2) = 5 m.

Question 2

If tan θ = 1, what is the value of θ in degrees (where 0° ≤ θ ≤ 90°)?

Accepted Answer

tan 45° = 1 is a standard trigonometric ratio for a 45° angle in a 45-45-90 triangle.

Question 3

In a right-angled triangle, if sin θ = 3/5, what is the value of cos θ?

Accepted Answer

Using the identity sin²θ + cos²θ = 1: (3/5)² + cos²θ = 1 gives cos²θ = 16/25, so cos θ = 4/5.

Question 4

Which angle is complementary to 28°?

Accepted Answer

Two angles are complementary when their sum equals 90°; therefore 90° − 28° = 62°.

Question 5

If sin θ = 3/5, find the value of cos θ using the Pythagorean identity sin²θ + cos²θ = 1, where θ is acute.

Accepted Answer

Using sin²θ + cos²θ = 1: (3/5)² + cos²θ = 1 gives cos²θ = 16/25, so cos θ = 4/5 for acute angle.

Question 6

Simplify: sec²θ − tan²θ

Accepted Answer

The identity sec²θ − tan²θ = 1 is a standard trigonometric identity derived from sin²θ + cos²θ = 1.

Question 7

If tan θ = 1, find sin θ + cos θ using trigonometric identities, where θ is acute.

Accepted Answer

tan θ = 1 means θ = 45°, so sin 45° = cos 45° = 1/√2, thus sin θ + cos θ = 2/√2 = √2.

Question 8

If tan(42°) = 0.9, what is cot(48°)?

Accepted Answer

Since 42° and 48° are complementary angles (42° + 48° = 90°), tan(42°) = cot(48°) by the complementary angle property.

Question 9

A man stands 30 m away from the base of a tower. The angle of elevation to the top of the tower is 60°. What is the height of the tower? (Use √3 ≈ 1.73)

Accepted Answer

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

Question 10

From the top of a building 20 m high, the angle of depression to a point on the ground is 45°. What is the horizontal distance from the building to that point?

Accepted Answer

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

Question 11

A ladder leans against a wall such that its foot is 5 m away from the wall and makes an angle of 60° with the ground. What is the length of the ladder?

Accepted Answer

Using cos(60°) = base/hypotenuse, we get length = 5/cos(60°) = 5/(1/2) = 10 m.

Question 12

If sin(35°) = 0.574, what is the value of cos(55°)?

Accepted Answer

Since 35° and 55° are complementary angles (35° + 55° = 90°), sin(35°) = cos(55°) by the complementary angle property.

Question 13

A man standing 40 m away from the base of a tower observes the angle of elevation to the top of the tower to be 30°. Find the height of the tower.

Accepted Answer

Using tan(30°) = height/distance, we get height = 40 × tan(30°) = 40 × (1/√3) = 40/√3 m.

Question 14

In a right-angled triangle, if sin θ = 3/5, what is the value of cos θ?

Accepted Answer

Using sin²θ + cos²θ = 1, we get cos²θ = 1 - (3/5)² = 1 - 9/25 = 16/25, so cos θ = 4/5.

Question 15

A ladder leans against a wall making an angle of 60° with the ground. If the ladder is 10 m long, how far is the base of the ladder from the wall?

Accepted Answer

The base distance is ladder × cos(60°) = 10 × 1/2 = 5 m.

Question 16

If tan θ = 7/24, find the value of sin θ.

Accepted Answer

In a right triangle with tan θ = 7/24, the hypotenuse is √(7² + 24²) = 25, so sin θ = opposite/hypotenuse = 7/25.

Question 17

From a point on the ground 30 m away from the base of a tower, the angle of elevation to the top is 45°. What is the height of the tower?

Accepted Answer

Height = distance × tan(45°) = 30 × 1 = 30 m.

Question 18

A ladder leans against a wall such that its foot is 6 m away from the wall and the ladder makes an angle of 60° with the ground. What is the length of the ladder?

Accepted Answer

Using cos(60°) = base/hypotenuse, we get hypotenuse = 6/cos(60°) = 6/(1/2) = 12 m.

Question 19

From the top of a cliff 80 m high, the angle of depression to a boat on the water is 45°. How far is the boat from the base of the cliff?

Accepted Answer

Using tan(45°) = height/distance, we get distance = 80/tan(45°) = 80/1 = 80 m.

Question 20

A tree casts a shadow of 15 m on the ground when the angle of elevation of the sun is 60°. What is the height of the tree?

Accepted Answer

Using tan(60°) = height/shadow, we get height = 15 × tan(60°) = 15 × √3 = 15√3 m.

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