Lesson 15 ~25 min Unit 3 · Trigonometry +85 XP

Combined Elevation and Depression

Multi-step problems with two angles in the same diagram. Often you must split the situation into two right triangles or use the elevation-depression equality.

Today's hook: From the top of a cliff you see a boat at 20° depression and the horizon at almost-zero depression. Combined with cliff height, can you estimate boat distance accurately?

0/5QUESTS

Printable Worksheets

Print or save as PDF — or build a custom worksheet from any module's questions.

Build Apply Master Build custom

Think First

warm-up

Two angles appear in one diagram (e.g. an elevation AND a depression). Will both angles use the same triangle, or different triangles? Sketch a scenario where both appear.

Record your answer in your workbook.

The Big Idea

+5 XP

Combined elevation/depression problems use TWO triangles that often share a common side. The trick is to label both clearly and use one ratio per triangle.

Common setup: a tall observer (or a high vantage point) sees two objects — one higher up (elevation) and one lower down (depression). Each angle has its own triangle with its own opposite and adjacent. Often the horizontal distance is shared between the two triangles. Set up TWO trig equations, solve them as a pair.

Two angles, two triangles — often shared side

Sketch BOTH triangles

Each angle in its own labelled triangle.

Shared side

Identify which side is common to both triangles — usually the horizontal.

Two equations

Set up one trig equation per triangle; combine if necessary.

What You'll Master

objectives

Know

Two angles can appear in the same physical setup
Each angle defines its own right triangle
The horizontal distance is often common between two triangles

Understand

How to identify shared sides between triangles
Why setting up two equations sometimes lets you solve for an unknown not given
When the elevation-depression equality saves work

Can Do

Solve problems with two angles
Use one angle to find an intermediate length, then the other angle
Cross-check with sense (distances should be positive and reasonable)

Words You Need

vocabulary

Combined problemA scenario with more than one angle of elevation or depression in the same setup.

Shared sideA side that belongs to both triangles in a combined problem.

Two-equation methodSetting up one trig equation per triangle and solving them together.

Vantage pointThe high (or low) position from which both angles are measured.

Total heightSometimes the answer requires adding heights from two triangles.

Horizontal axisThe shared horizontal line connecting the two triangles when they share an adjacent side.

Spot the Trap

heads-up

Wrong: Trying to fit both angles into one triangle — rarely possible.

Right: Draw two separate right triangles; identify the common side.

Wrong: Adding angles directly: 20° + 5° = 25° — not a meaningful side measurement.

Right: Each angle gives a SEPARATE side ratio. Don't combine angles arithmetically.

Method for Two-Angle Problems

+5 XP

A reliable five-step plan:

1. Sketch the whole scenario in one big diagram. 2. Identify the TWO right triangles. 3. Label each triangle's opp, adj, hyp and angle. 4. Spot the shared side. 5. Write one trig equation per triangle, then combine to solve.

Two triangles, two equations, often a shared side

Big sketch

One diagram with both angles visible saves confusion.

Track shared length

Often the shared side is the ‘link’ that solves the problem.

Tan x 2

Most combined problems use tan twice (once per triangle).

Worked Pattern: Two Boats

+5 XP

Classic example: a cliff-top observer sees two boats at different depressions on the same straight line. Find the gap between the boats.

Step	Action
1	Cliff height $h$ is the shared opposite.
2	Distance to near boat = $h/\tan(\theta_1)$
3	Distance to far boat = $h/\tan(\theta_2)$
4	Gap = far − near

Gap = $h \bigl(\dfrac{1}{\tan\theta_2} - \dfrac{1}{\tan\theta_1}\bigr)$

Smaller angle = further

As depression decreases, the boat is further away.

Same height = key link

The cliff height is the shared opp in both triangles.

Subtract not add

Gap = (far horizontal) - (near horizontal).

Watch Me Solve It · 3 examples

Watch Me Solve It · Cliff with two boats

+15 XP per step

PROBLEM

From a 50 m cliff, boat A is at depression 30° and boat B is at depression 18°. Both boats are on the same straight line. Find the distance between them (2 d.p.).

1

Distance to A

$d_A = 50/\tan 30° \approx 86.60$ m
2

Distance to B

$d_B = 50/\tan 18° \approx 153.88$ m
3

Gap

$d_B - d_A \approx 67.28$ m

Answer$\approx 67.28$ m apart

Watch Me Solve It · Observer sees both above and below

+15 XP per step

PROBLEM

From a balcony 10 m above ground, you see a bird at elevation 20° and a dog at depression 35°. Find the total vertical distance between the bird and the dog (assume same horizontal distance from you).

1

Same horizontal

Let $d$ = horizontal distance. Suppose $d = 8$ m.
2

Bird above eye

$h_{bird} = d\tan 20° \approx 2.91$ m
3

Dog below eye

$h_{dog} = d\tan 35° \approx 5.60$ m. Total $= 2.91+5.60 \approx 8.52$ m apart.

Answer$\approx 8.52$ m (for $d = 8$ m)

Watch Me Solve It · Tower at two distances

+15 XP per step

PROBLEM

From point A, a tower top is seen at elevation 25°. Moving 30 m closer (to point B), the elevation is 40°. Find the tower height (2 d.p.).

1

Two equations

$h = d_B \tan 40°$ and $h = (d_B + 30)\tan 25°$
2

Equate

$d_B \tan 40° = (d_B + 30)\tan 25°$
3

Solve

$d_B (\tan 40° - \tan 25°) = 30\tan 25°$. $d_B \approx 37.62$ m. $h \approx 31.57$ m.

Answer$h \approx 31.57$ m

Common Pitfalls

heads-up

One-triangle thinking

Trying to cram both angles into the same triangle.

Fix: Draw TWO right triangles, even if they overlap on a shared side.

Adding angles

Treating 20° + 35° as the total angle.

Fix: Each angle defines its own ratio — not arithmetic.

Confused units / sense

Distances that are negative or absurdly large.

Fix: Sense-check: smaller depression = further away; bigger elevation = steeper.

Copy Into Your Books

Two triangles

Each angle = own triangle
Same opp or same adj often shared
Set up TWO trig equations

Shared side trick

Same height + different angles
$h = d\tan\theta$ for each

Solve

Subtract for gap
Equate for tower height
Add for total drop

Sense check

Smaller angle = larger distance
Distances positive
Angles 0 to 90

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Two-Angle Drills

4 problems

Four quick drills to lock in today's skill. Try each, then reveal the answer.

1 From 100 m cliff: boat A at depression 45°, boat B at 30°. Distance between (2 d.p.)?

$100/\tan 30° - 100/\tan 45° \approx 173.21 - 100 \approx 73.21$.$\approx 73.21$ m
2 Tower elevation 60° from 20 m away. Height?

$h = 20\tan 60° \approx 34.64$.$\approx 34.64$ m
3 From 80 m cliff: depression 35° (boat). Distance to boat (2 d.p.)?

$80/\tan 35° \approx 114.25$.$\approx 114.25$ m
4 From eye height 1.7 m, a wall is 8 m away. Top of wall is at elevation 25°. Wall height above ground (2 d.p.)?

$8\tan 25° + 1.7 \approx 3.73 + 1.7 \approx 5.43$.$\approx 5.43$ m

Complete in your workbook.

Quick Check · 5 questions

Combined elevation/depression problems usually require:

+10 XP

From a 100 m cliff, a boat is at depression 25°. Horizontal distance to boat (2 d.p.):

+10 XP

Two cars are seen from a 30 m bridge at depressions 50° and 25° (same line). Distance between cars (2 d.p.):

+10 XP

From point P, tower elevation is 25°. Moving 40 m closer makes elevation 40°. The shared side in both triangles is:

+10 XP

You can swap a depression angle for an elevation angle when:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. From a 30 m lighthouse, a yacht is at angle of depression 18° and a kayak is at depression 36°, both on the same line directly out to sea. (a) Find the horizontal distance to each. (b) Find the distance between them.

Answer in your workbook.

ApplyMedium3 MARKS

Q7. From a balcony 20 m above the ground, you see a bird at angle of elevation 25° and a dog directly below the bird at angle of depression 40°. Find the total vertical distance from bird to dog (2 d.p.). [Assume both are at the same horizontal distance.]

Answer in your workbook.

ReasonHard4 MARKS

Q8. From point A on flat ground, the elevation to a tower top is 30°. From point B (50 m closer to the tower along the same line), the elevation is 50°. Find the tower height (2 d.p.).

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — Two triangles.

2. D — $100/\tan 25° \approx 214.45$.

3. A — $30(1/\tan 25 - 1/\tan 50) \approx 39.16$.

4. C — Tower height (opp).

5. B — Always (alternate angles).

Show Your Working Model Answers

Q6 (3 marks): (a) Yacht: $30/\tan 18° \approx 92.33$ m. Kayak: $30/\tan 36° \approx 41.29$ m [2]. (b) Gap $\approx 51.04$ m [1].

Q7 (3 marks): Let $d$ = horizontal distance to bird and dog. Bird above eye: $d\tan 25°$. Dog below eye: $d\tan 40°$ [1]. Total separation = bird-height-above-eye + dog-depth-below-eye = $d(\tan 25° + \tan 40°) = d \times 1.305$ [1]. If $d=10$ m, total $\approx 13.05$ m [1].

Q8 (4 marks): Let $d$ = distance from B to base. $h = d\tan 50°$ and $h = (d+50)\tan 30°$ [1]. Equate: $d\tan 50° = (d+50)\tan 30°$. $d(\tan 50° - \tan 30°) = 50\tan 30°$ [1]. $d = 50\tan 30°/(\tan 50° - \tan 30°) \approx 28.87/0.614 \approx 47.04$ m [1]. $h \approx 47.04\tan 50° \approx 56.05$ m [1].

Stretch Challenge · +25 XP, +10 coins

Climber and helicopter

A climber on a cliff at 80 m above the valley floor sees a helicopter at elevation 25° (above eye level). A spotter on the valley floor sees the same helicopter at elevation 50°. The helicopter, climber and spotter are on the same vertical plane. Find the height of the helicopter above the valley floor (2 d.p.).

Reveal solution

Let $h$ = helicopter height above valley floor, $d$ = horizontal distance from spotter (=climber's horizontal) to helicopter. Spotter: $h = d\tan 50°$. Climber: $h - 80 = d\tan 25°$. Subtract: $80 = d(\tan 50° - \tan 25°)$, $d \approx 80/0.7253 \approx 110.30$ m. $h \approx 110.30\tan 50° \approx 131.43$ m.

Quick Review

Two triangles

One per angle

Shared side

Often horizontal or vertical

Two equations

Solve simultaneously

Most use tan

Height + ground distances

Depression = elevation

Alternate angles

Sense check

Distances positive and reasonable

Your Badges

0 of 6

First Steps

3-Day Streak

3 in a Row

Lesson Ace

Stretch Seeker

Daily Warrior

Mark lesson as complete

Tick when you've finished Learn, Practice and the Stretch. Earns +85 XP and +25 coins.

Unit overview · Maths subject page