M
Math Coach Amy
Amy Ferguson Moncure

IB AA · Unit 2 · 5C Applications of Trigonometry

Elevation and Depression

Use a right triangle for an angle of elevation or depression. Two angles give two equations.

Jump to section:

Key Ideas Easy Medium Hard

tan θ = oppositeadjacent

What to Remember

Elevation and depression

An angle of elevation looks up from the horizontal.

An angle of depression looks down from the horizontal.

The right triangle uses the same opposite and adjacent either way.

One triangle

tan θ = oppositeadjacent

Add an eye height or a person’s height only after you have found the height above that eye.

Two angles

When a person moves closer, write a tangent equation from each place.

The difference of the two horizontal distances is the distance walked.

Easy

One angle

Find the height above the eye first. Add the eye height on the next lines.

Example
From a point on level ground 30 m from a tower, the angle of elevation of the top is 28°. The eye is 1.6 m above the ground. Find the height of the tower.

tan 28° = height above eyedistance

height above eye = 30 × tan 28°

height above eye = 16.0 m

Total height = 16.0 + 1.60

Total height = 17.6 m

Answer: 17.6 m

1
From a point 45 m from the base, the angle of elevation of the top of a building is 22°. Find the height.
Show Answer

tan 22° = height above eyedistance

height above eye = 45 × tan 22°

height above eye = 18.2 m

Answer: 18.2 m

2
From a point 18 m from the base, the angle of elevation of the top of a tree is 35°. The eye is 1.70 m above the ground. Find the height.
Show Answer

tan 35° = height above eyedistance

height above eye = 18 × tan 35°

height above eye = 12.6 m

Total height = 12.6 + 1.70

Total height = 14.3 m

Answer: 14.3 m

3
From the top of a cliff 36 m above the sea, the angle of depression of a boat is 12°. Find the distance from the boat to the base of the cliff.
Show Answer

tan 12° = 36distance

distance = 36tan 12°

distance = 169 m

Answer: 169 m

4
From a point 25 m from the base, the angle of elevation of the top of a flagpole is 41°. Find the height.
Show Answer

tan 41° = height above eyedistance

height above eye = 25 × tan 41°

height above eye = 21.7 m

Answer: 21.7 m

5
From a point 80 m from the base, the angle of elevation of the top of a monument is 8°. The eye is 2 m above the ground. Find the height.
Show Answer

tan 8° = height above eyedistance

height above eye = 80 × tan 8°

height above eye = 11.2 m

Total height = 11.2 + 2

Total height = 13.2 m

Answer: 13.2 m

Medium

Two people, or a window

Each angle gets its own tangent equation.

Example
Ava stands 20 m from a tower and sees the top at an elevation of 32°. Ben stands 28 m farther from the tower, on the same straight line. Find the angle of elevation from Ben to the top.

tan 32° = h20

h = 20 × tan 32°

h = 12.5 m

Farther distance = 20 + 28 = 48 m

tan θ = 12.548

θ = 14.6°

Answer: 14.6°

1
Ava stands 15 m from a tower and sees the top at an elevation of 27°. Ben stands 22 m farther from the tower, on the same straight line. Find the angle of elevation from Ben to the top.
Show Answer

tan 27° = h15

h = 15 × tan 27°

h = 7.64 m

Farther distance = 15 + 22 = 37 m

tan θ = 7.6437

θ = 11.7°

Answer: 11.7°

2
Ava stands 12 m from a tower and sees the top at an elevation of 24°. Ben stands 18 m farther from the tower, on the same straight line. Find the angle of elevation from Ben to the top.
Show Answer

tan 24° = h12

h = 12 × tan 24°

h = 5.34 m

Farther distance = 12 + 18 = 30 m

tan θ = 5.3430

θ = 10.1°

Answer: 10.1°

3
A visitor sits 3.20 m from a wall. The bottom of a painting is at an elevation of 28° and the top is at an elevation of 47°. Find the height of the painting.
Show Answer

Height of the bottom above eye level = 3.20 × tan 28°

Bottom = 1.70 m

Height of the top above eye level = 3.20 × tan 47°

Top = 3.43 m

Painting = 3.43 − 1.70

Painting = 1.73 m

Answer: 1.73 m

4
A visitor sits 2.50 m from a wall. The bottom of a painting is at an elevation of 33° and the top is at an elevation of 58°. Find the height of the painting.
Show Answer

Height of the bottom above eye level = 2.50 × tan 33°

Bottom = 1.62 m

Height of the top above eye level = 2.50 × tan 58°

Top = 4.00 m

Painting = 4.00 − 1.62

Painting = 2.38 m

Answer: 2.38 m

5
From a window 11 m above the ground, the angle of depression of a car is 16°. Find the distance from the car to the base of the building.
Show Answer

tan 16° = 11distance

distance = 11tan 16°

distance = 38.4 m

Answer: 38.4 m

Hard

Walk closer, or two objects

The distance walked is the difference of the two adjacent sides.

Example
The angle of elevation of the top of a statue is 18°. After walking 20 m straight toward the statue, the angle of elevation is 32°. Find the height of the statue above eye level.

tan 18° = hx

x = htan 18°

tan 32° = hx − 20

x − 20 = htan 32°

h × 1tan 18° − h × 1tan 32° = 20

h = 201tan 18° − 1tan 32°

h = 13.5 m

Answer: 13.5 m

1
The angle of elevation of the top of a statue is 21°. After walking 12 m straight toward the statue, the angle of elevation is 38°. Find the height of the statue above eye level.
Show Answer

tan 21° = hx

x = htan 21°

tan 38° = hx − 12

x − 12 = htan 38°

h × 1tan 21° − h × 1tan 38° = 12

h = 121tan 21° − 1tan 38°

h = 9.06 m

Answer: 9.06 m

2
The angle of elevation of the top of a statue is 24°. After walking 8 m straight toward the statue, the angle of elevation is 45°. Find the height of the statue above eye level.
Show Answer

tan 24° = hx

x = htan 24°

tan 45° = hx − 8

x − 8 = htan 45°

h × 1tan 24° − h × 1tan 45° = 8

h = 81tan 24° − 1tan 45°

h = 6.42 m

Answer: 6.42 m

3
From a lighthouse, two buoys lie on the same straight line south of the lamp. The buoys are 16 m apart. The angles of depression are 38° to the nearer buoy and 31° to the farther buoy. Find the height of the lamp above the water.
Show Answer

Distance to the nearer buoy = htan 38°

Distance to the farther buoy = htan 31°

Difference = 16

h = 161tan 31° − 1tan 38°

h = 41.6 m

Answer: 41.6 m

4
A visitor sits 4.10 m from a wall. The bottom of a painting is at an elevation of 26° and the top is at an elevation of 61°. Find the height of the painting.
Show Answer

Height of the bottom above eye level = 4.10 × tan 26°

Bottom = 2.00 m

Height of the top above eye level = 4.10 × tan 61°

Top = 7.40 m

Painting = 7.40 − 2.00

Painting = 5.40 m

Answer: 5.40 m

5
From the top of a lighthouse, the angle of depression of a boat is 22°. The lighthouse lamp is 48 m above the water. Find the distance from the boat to the base of the lighthouse.
Show Answer

tan 22° = 48distance

distance = 48tan 22°

distance = 119 m

Answer: 119 m

Want to go over a solid or a triangle together?

Book a Session with Amy