Question 1

A man standing 30 metres away from the base of a tower observes the angle of elevation to the top of the tower as 60°. Find the height of the tower.

Accepted Answer

Step 1: Use the tangent ratio in the right triangle formed. tan(60°) = height / distance. Step 2: tan(60°) = √3, so √3 = height / 30. Step 3: height = 30√3 metres ≈ 51.96 metres. Therefore, the answer is 30√3 metres.

Question 2

From a point on the ground, the angle of elevation to the top of a 20-metre tall building is 45°. How far is the point from the base of the building?

Accepted Answer

Step 1: Use tan(45°) = height / distance. Step 2: tan(45°) = 1, so 1 = 20 / distance. Step 3: distance = 20 metres. Therefore, the answer is 20 metres.

Question 3

If tan(x) = cot(40°), then x = ?

Accepted Answer

Step 1: Use the complementary angle property: cot(θ) = tan(90° - θ). Step 2: Therefore, cot(40°) = tan(90° - 40°) = tan(50°). Step 3: Since tan(x) = cot(40°) = tan(50°), we have x = 50°.

Question 4

An observer on the ground sees the top of a tree at an angle of elevation of 60°. If the observer is 15 metres away from the base of the tree, what is the height of the tree? (Use √3 ≈ 1.73)

Accepted Answer

Step 1: tan(60°) = height / distance. Step 2: tan(60°) = √3, so √3 = height / 15. Step 3: height = 15√3 = 15 × 1.73 = 25.95 metres ≈ 26 metres. Therefore, the answer is 15√3 metres or approximately 26 metres.

Question 5

From the top of a 48-metre tall building, the angle of depression to a point on the ground is 30°. What is the horizontal distance from the building to that point?

Accepted Answer

Step 1: The angle of depression equals the angle of elevation from the ground point. tan(30°) = height / distance. Step 2: tan(30°) = 1/√3, so 1/√3 = 48 / distance. Step 3: distance = 48√3 metres. Therefore, the answer is 48√3 metres.

Question 6

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

Accepted Answer

Step 1: Recognize that 35° and 55° are complementary angles (35° + 55° = 90°). Step 2: Apply the complementary angle property: sin(θ) = cos(90° - θ). Step 3: Therefore, sin(35°) = cos(90° - 35°) = cos(55°). Step 4: Since sin(35°) = k, we have cos(55°) = k.

Question 7

sin²(17°) + sin²(73°) = ?

Accepted Answer

Step 1: Observe that 17° + 73° = 90°, so 73° = 90° - 17°. Step 2: Apply the complementary angle property: sin(90° - θ) = cos(θ), so sin(73°) = cos(17°). Step 3: Substitute: sin²(17°) + sin²(73°) = sin²(17°) + cos²(17°). Step 4: Apply the Pythagorean identity: sin²(17°) + cos²(17°) = 1.

Question 8

If cot(42°) = m, then sec(48°) × sin(48°) = ?

Accepted Answer

Step 1: Recognize that 42° + 48° = 90°, so they are complementary angles. Step 2: cot(42°) = tan(48°), so m = tan(48°). Step 3: sec(48°) × sin(48°) = (1/cos(48°)) × sin(48°) = sin(48°)/cos(48°) = tan(48°). Step 4: Therefore, sec(48°) × sin(48°) = m.

Question 9

tan(28°) × tan(62°) = ?

Accepted Answer

Step 1: Observe that 28° + 62° = 90°, so these are complementary angles. Step 2: For complementary angles, tan(θ) × tan(90° - θ) = tan(θ) × cot(θ) = 1. Step 3: Therefore, tan(28°) × tan(62°) = 1.

Question 10

A ladder leans against a wall such that it makes an angle of 30° with the ground. If the ladder is 10 metres long, what is the height at which the ladder touches the wall?

Accepted Answer

Step 1: The ladder forms the hypotenuse of a right triangle. sin(30°) = height / hypotenuse. Step 2: sin(30°) = 0.5, so height = 10 × 0.5 = 5 metres. Therefore, the answer is 5 metres.

Question 11

In a right triangle, if cot θ = 8/15, find the value of sin θ.

Accepted Answer

Step 1: Given cot θ = 8/15, so adjacent/opposite = 8/15, which means adjacent = 8, opposite = 15. Step 2: Using Pythagoras: hypotenuse² = 8² + 15² = 64 + 225 = 289, so hypotenuse = 17. Step 3: sin θ = opposite/hypotenuse = 15/17.

Question 12

Two buildings are 60 m apart. From the top of the first building (height 30 m), the angle of elevation to the top of the second building is 45°. Find the height of the second building.

Accepted Answer

Step 1: Distance between buildings = 60 m. Height of first building = 30 m. Angle of elevation from top of first to top of second = 45°. Step 2: Let height of second building = h. Vertical difference = (h - 30). Step 3: Using tan(45°) = (h - 30)/60, and tan(45°) = 1, we get h - 30 = 60. Step 4: h = 90 m.

Question 13

A person standing on the ground observes the angle of elevation to the top of a tree as 45°. After walking 10 m towards the tree, the angle of elevation becomes 60°. Find the height of the tree. [Use √3 ≈ 1.732]

Accepted Answer

Step 1: Let height = h, initial distance = d. From first position: tan(45°) = h/d, so h = d (since tan(45°) = 1). Step 2: From second position (10 m closer): tan(60°) = h/(d-10), so h = (d-10)√3. Step 3: Equating: d = (d-10)√3. Step 4: d = d√3 - 10√3. Step 5: d(√3 - 1) = 10√3. Step 6: d = 10√3/(√3 - 1) = 10√3(√3 + 1)/[(√3 - 1)(√3 + 1)] = 10√3(√3 + 1)/2 = 10(3 + √3)/2 = 5(3 + √3) = 15 + 5√3 ≈ 23.66 m. Step 7: h = d ≈ 23.66 m.

Question 14

A man standing 40 m away from the base of a tower observes the angle of elevation to the top of the tower as 60°. Find the height of the tower (in metres). [Use √3 ≈ 1.732]

Accepted Answer

Step 1: Let height of tower = h. Distance from base = 40 m. Angle of elevation = 60°. Step 2: Using tan(60°) = h/40, we get h = 40 × tan(60°) = 40 × √3. Step 3: h = 40 × 1.732 = 69.28 m ≈ 69.3 m.

Question 15

From the top of a cliff 100 m high, the angle of depression to a boat in the sea is 30°. How far is the boat from the base of the cliff? [Use √3 ≈ 1.732]

Accepted Answer

Step 1: Height of cliff = 100 m. Angle of depression = 30°. Step 2: Angle of depression equals angle of elevation from boat's perspective. Using tan(30°) = 100/distance, we get distance = 100/tan(30°). Step 3: tan(30°) = 1/√3. Distance = 100 × √3 = 100 × 1.732 = 173.2 m.

Question 16

In a right-angled triangle, if sin θ = 3/5, then find the value of tan θ.

Accepted Answer

Step 1: Given sin θ = 3/5, which means opposite/hypotenuse = 3/5. Step 2: Using Pythagoras theorem: adjacent² = hypotenuse² - opposite² = 5² - 3² = 25 - 9 = 16, so adjacent = 4. Step 3: tan θ = opposite/adjacent = 3/4.

Question 17

If cos θ = 7/25, find the value of (sin θ + tan θ).

Accepted Answer

Step 1: Given cos θ = 7/25, so adjacent = 7, hypotenuse = 25. Step 2: Using Pythagoras: opposite² = 25² - 7² = 625 - 49 = 576, so opposite = 24. Step 3: sin θ = 24/25 and tan θ = 24/7. Step 4: sin θ + tan θ = 24/25 + 24/7 = (24×7 + 24×25)/(25×7) = (168 + 600)/175 = 768/175.

Question 18

In a right triangle, if tan θ = 5/12, what is the value of sec θ?

Accepted Answer

Step 1: Given tan θ = 5/12, so opposite = 5, adjacent = 12. Step 2: Using Pythagoras: hypotenuse² = 5² + 12² = 25 + 144 = 169, so hypotenuse = 13. Step 3: sec θ = hypotenuse/adjacent = 13/12.

Question 19

If cos θ = 5/13 and sin θ > 0, find the value of (tan θ × sec θ).

Accepted Answer

Step 1: Given cos θ = 5/13, so adjacent = 5, hypotenuse = 13. Step 2: Using Pythagoras: opposite² = 13² - 5² = 169 - 25 = 144, so opposite = 12 (positive since sin θ > 0). Step 3: tan θ = 12/5 and sec θ = 13/5. Step 4: tan θ × sec θ = (12/5) × (13/5) = 156/25.

Question 20

If tan(62°) × tan(28°) = m, what is the value of m?

Accepted Answer

Step 1: Observe that 62° and 28° are complementary angles (62° + 28° = 90°). Step 2: Apply the complementary angle identity: tan(θ) = cot(90° − θ). Step 3: Therefore, tan(62°) = cot(28°) = 1/tan(28°). Step 4: Substitute: m = tan(62°) × tan(28°) = [1/tan(28°)] × tan(28°) = 1.

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