Some Applications of Trigonometry — Practice

Question 1

From the top of a tower the angle of depression of a car is 30°. What is the angle of elevation of the tower's top from the car?

Accepted Answer

Step 1 — Depression from top = elevation from ground (alternate angles).

✦ Answer: 30°.

Question 2

A ladder leans against a wall making 60° with the ground; its foot is 2 m from the wall. How long is the ladder?

Accepted Answer

Step 1 — cos60° = adjacent/hyp = 2/L ⇒ 1/2 = 2/L.

Step 2 — L = 4 m.

✦ Answer: 4 m.

Question 3

The shadow of a pole is √3 times its height. Find the sun's angle of elevation.

Accepted Answer

Step 1 — tan θ = height/shadow = h/(√3 h) = 1/√3.

Step 2 — θ = 30°.

✦ Answer: 30°.

Question 4

The angle of elevation of the top of a tower from a point 30 m away is 60°. Find the tower's height.

Accepted Answer

Step 1 — tan60° = h/30 ⇒ √3 = h/30.

Step 2 — h = 30√3 ≈ 51.96 m.

✦ Answer: 30√3 m ≈ 51.96 m.

Question 5

From a point the angle of elevation to a cliff top of height 100 m is 45°. How far is the point from the cliff base?

Accepted Answer

Step 1 — tan45° = 100/d ⇒ 1 = 100/d.

Step 2 — d = 100 m.

✦ Answer: 100 m.

Question 6

A 1.5 m tall boy stands 15 m from a tower. The elevation of the top from his eyes is 45°. Find the tower's height.

Accepted Answer

Step 1 — tan45° = (H − 1.5)/15 ⇒ 1 = (H − 1.5)/15.

Step 2 — H − 1.5 = 15 ⇒ H = 16.5 m.

✦ Answer: 16.5 m.

Question 7

The angle of elevation of a tower's top from a point is 30°. Moving 20 m towards the tower, it becomes 60°. Find the tower's height.

Accepted Answer

Step 1 — Let height h, nearer distance x. tan60° = h/x ⇒ h = x√3. tan30° = h/(x+20) ⇒ h = (x+20)/√3.

Step 2 — x√3 = (x+20)/√3 ⇒ 3x = x + 20 ⇒ x = 10.

Step 3 — h = 10√3 ≈ 17.32 m.

✦ Answer: 10√3 m ≈ 17.32 m.

Question 8

From the top of a 50 m building, the angles of depression of the top and bottom of a tower are 30° and 60°. Find the tower's height.

Accepted Answer

Step 1 — Let horizontal distance d. To the bottom: tan60° = 50/d ⇒ d = 50/√3.

Step 2 — To the top of the tower (height h): the vertical drop is (50 − h); tan30° = (50 − h)/d ⇒ (50 − h) = d/√3 = 50/3.

Step 3 — h = 50 − 50/3 = 100/3 ≈ 33.33 m.

✦ Answer: 100/3 m ≈ 33.33 m.

Question 9

A tower is 20 m high. From its foot the angle of elevation of a building's top is 60°, and from the building's foot the elevation of the tower's top is 30°. Find the building's height.

Accepted Answer

Step 1 — Let distance between them d, building height H. From tower foot: tan60° = H/d ⇒ H = d√3.

Step 2 — From building foot: tan30° = 20/d ⇒ d = 20√3.

Step 3 — H = 20√3 · √3 = 60 m.

✦ Answer: 60 m.

Question 10

The angle of elevation of the top of a 75 m lighthouse from a boat is 30°. How far is the boat from the lighthouse?

Accepted Answer

Step 1 — tan30° = 75/d ⇒ 1/√3 = 75/d.

Step 2 — d = 75√3 ≈ 129.9 m.

✦ Answer: 75√3 m ≈ 129.9 m.

Practice — Some Applications of Trigonometry

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