Lesson 14 ~25 min Unit 3 · Trigonometry +85 XP

Angles of Depression

Measure how steeply you look DOWN to see a low point from a higher position. The horizontal at your eye is the adjacent; the vertical drop is the opposite. Use the alternate-angles property to swap depression for elevation.

Today's hook: A lifeguard on a 12 m tower spots a swimmer in the water at an angle of depression of 28°. How far is the swimmer from the base of the tower?

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Think First

warm-up

You stand on a 10 m cliff and look down at a kayaker. The angle of depression (from horizontal looking DOWN) is 30°. Without computing, would the kayaker BE the same triangle as if YOU were the kayaker looking UP at 30°? Sketch both views.

Record your answer in your workbook.

The Big Idea

+5 XP

The angle of depression is measured from the horizontal looking DOWN at an object below you. Like elevation, the angle is taken from the flat horizontal line at your eye — never from the vertical.

In the right triangle: the horizontal at your eye is the adjacent; the vertical drop is the opposite; the line of sight down is the hypotenuse. Crucially, by ALTERNATE INTERIOR ANGLES, the angle of depression FROM your eye equals the angle of elevation FROM the object below.

Angle of depression: from horizontal looking DOWN

From horizontal

At your eye, draw a flat line; the angle drops downward from that.

Equals elevation

By alternate angles, depression FROM you = elevation FROM the object.

Looking down

You are higher than the object. The line of sight slants downward.

What You'll Master

objectives

Know

Angle of depression is measured from the horizontal looking DOWN
Depression from above = elevation from below (alternate angles)
Side labels: horizontal at eye = adj; vertical drop = opp; sight line = hyp

Understand

Why depression and elevation are equal (parallel lines, transversal, alternate interior angles)
How to set up a triangle correctly for a depression problem
When to use which trig ratio

Can Do

Sketch a depression scenario
Find horizontal distance or vertical drop
Swap to elevation when convenient

Words You Need

vocabulary

Angle of depressionThe angle between the horizontal at the observer's eye and the line of sight going DOWN to an object.

Alternate interior anglesEqual angles formed when a line crosses two parallel lines on opposite sides.

Parallel horizontalsAt any two heights, the horizontal lines are parallel — that's why depression = elevation.

Observer at heightThe person looking down. Their eye is the vertex of the depression angle.

TransversalA line crossing two parallel lines — here, the line of sight.

Depression = elevationThe depression angle from above and the elevation angle from below are equal (alternate angles).

Spot the Trap

heads-up

Wrong: Measuring the depression from the vertical instead of horizontal — off by 90°.

Right: Depression measured at the observer's eye from the flat horizontal line, sloping downward.

Wrong: Placing the depression angle at the wrong vertex of the triangle (e.g. at the object below).

Right: Use the depression-equals-elevation rule to redraw the triangle with the angle at the object — it's often easier.

Depression = Elevation

+5 XP

At parallel horizontal lines, depression (from above) and elevation (from below) are ALTERNATE INTERIOR ANGLES, so they are equal.

Picture two parallel horizontals — one at the observer's eye, one at the object below. The line of sight is a transversal cutting both. The depression angle (above) and the elevation angle (below) are alternate interior angles — they are equal. You can solve depression problems by re-imagining them as elevation problems from below.

Depression (at top) = Elevation (at bottom) by alternate angles

Equal angles

Same number for both — just re-draw to make it elevation.

Pick easier vertex

Place the angle wherever the calculation is simplest.

Hyp is hyp

The line of sight (slanted) is always the hypotenuse, regardless of viewpoint.

Setting Up a Depression Problem

+5 XP

Five-step method: sketch with the horizontal at observer's eye, mark depression, label sides, choose ratio, solve.

Want	Use
Horizontal distance from below object	$\tan\theta = $ vertical drop / horizontal → rearrange
Vertical drop	$\tan\theta = $ drop / horizontal → multiply
Slant distance (line of sight)	Use sin or cos

Most depression problems use tan (height + ground distance)

Eye height counts

If asked ‘how far from base of cliff’, the height above water = cliff height + eye height (or set up the triangle from eye).

Tan dominates

Most depression problems compare a height (opp) and a ground distance (adj).

Re-label for elevation

If it's easier, redraw with the angle at the object below — same triangle.

Watch Me Solve It · 3 examples

Watch Me Solve It · Lifeguard spots swimmer

+15 XP per step

PROBLEM

A lifeguard on a 12 m tower sees a swimmer at angle of depression 28°. How far is the swimmer from the base of the tower (2 d.p.)?

1

Set up triangle

opp = 12 m (drop), adj = horizontal distance, $\theta = 28°$
2

Use tan

$\tan 28° = 12/$adj
3

Compute

adj $= 12/\tan 28° \approx 12/0.5317 \approx 22.57$ m

Answer$\approx 22.57$ m from base

Watch Me Solve It · Find depression angle

+15 XP per step

PROBLEM

A ship is 200 m from the base of a 50 m cliff. Find the angle of depression of the ship from the top of the cliff (1 d.p.).

1

Use tan

$\tan\theta = 50/200 = 0.25$

Drop / horizontal.
2

Inverse

$\theta = \tan^{-1}(0.25) \approx 14.0°$
3

State

Angle of depression $\approx 14.0°$

Answer$\theta \approx 14.0°$

Watch Me Solve It · Aircraft above runway

+15 XP per step

PROBLEM

An aircraft at 1500 m altitude sights an airport at depression 8°. Find the horizontal distance to the airport (to the nearest metre).

1

Set up

opp = 1500, $\theta = 8°$, adj = ?
2

Apply

$\tan 8° = 1500/$adj
3

Compute

adj $= 1500/\tan 8° \approx 1500/0.1405 \approx 10676$ m

Answer$\approx 10676$ m

Common Pitfalls

heads-up

Wrong vertex

Drawing the depression angle at the object instead of the observer's eye.

Fix: Depression is at the OBSERVER, between horizontal and the downward sight line.

Missing the alternate-angles rule

Not realising depression from above equals elevation from below.

Fix: They are equal — use whichever is easier to set up.

Confusing depression with the vertical

Measuring the angle from the vertical (straight down) instead of horizontal.

Fix: Always from the horizontal — the angle is small for distant objects, near 90° only for nearly-overhead objects.

Copy Into Your Books

Angle of depression

From horizontal at eye, looking DOWN
Vertex at observer
0° (horizontal) to 90° (straight down)

= Elevation

Alternate interior angles
Two parallel horizontals
Same number

Triangle setup

Horizontal at eye = adj
Vertical drop = opp
Line of sight = hyp

Most use tan

Height + ground
No hyp needed
$\tan = $ drop / horizontal

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Depression Drills

4 problems

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

1 Tower 15 m, swimmer 25 m from base. Angle of depression (1 d.p.)?

$\tan^{-1}(15/25) \approx 31.0°$.$\approx 31.0°$
2 Tower 10 m, angle of depression 40°. Horizontal distance (2 d.p.)?

adj = $10/\tan 40° \approx 11.92$.$\approx 11.92$ m
3 Aircraft 1000 m up, depression 20°. Horizontal distance (nearest m)?

adj = $1000/\tan 20° \approx 2747$.$\approx 2747$ m
4 Depression from 50 m up = 35°. Vertical drop is 50 m. Horizontal distance (2 d.p.)?

adj $= 50/\tan 35° \approx 71.41$.$\approx 71.41$ m

Complete in your workbook.

Quick Check · 5 questions

The angle of depression is measured from:

+10 XP

A depression angle of 30° from the top of a cliff equals the angle of elevation from the foot. This is because of:

+10 XP

A 12 m tower views a swimmer at angle of depression 30°. Horizontal distance (2 d.p.):

+10 XP

From an aircraft 800 m up, the airport is at depression 12°. Slant distance (line of sight) to the airport (2 d.p.):

+10 XP

Ship 300 m from base of 80 m cliff. Depression angle (1 d.p.):

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. From the top of a 60 m lighthouse, a boat is observed at angle of depression 22°. (a) Find the horizontal distance from the boat to the base of the lighthouse (2 d.p.). (b) Find the slant distance from the boat to the top of the lighthouse (2 d.p.).

Answer in your workbook.

UnderstandEasy2 MARKS

Q7. Explain, using parallel lines, why the angle of depression of a boat from a cliff-top equals the angle of elevation of the cliff-top from the boat.

Answer in your workbook.

ReasonHard4 MARKS

Q8. A lighthouse is 50 m above sea level. Two ships are seen on the same line directly out to sea. Ship A is at depression 30°, ship B is at depression 18°. Find the distance between the two ships (2 d.p.).

Answer in your workbook.

Comprehensive Answers

Quick Check

1. A — From horizontal at eye, looking down.

2. C — Alternate interior angles.

3. B — $12/\tan 30° \approx 20.78$.

4. D — $800/\sin 12° \approx 3848.05$.

5. B — $\tan^{-1}(4/15) \approx 14.9°$.

Show Your Working Model Answers

Q6 (3 marks): (a) adj $= 60/\tan 22° \approx 148.49$ m [2]. (b) hyp $= 60/\sin 22° \approx 160.17$ m [1].

Q7 (2 marks): The horizontal line at the cliff-top and the horizontal line at the boat are parallel [1]. The line of sight is a transversal cutting both. The angle of depression at the top and the angle of elevation at the boat are alternate interior angles, so they are equal [1].

Q8 (4 marks): Distance to A $= 50/\tan 30° \approx 86.60$ m [1]. Distance to B $= 50/\tan 18° \approx 153.88$ m [1]. Ship B is further (smaller depression) [1]. Gap $= 153.88 - 86.60 = 67.28$ m [1].

Stretch Challenge · +25 XP, +10 coins

Helicopter on the move

A stationary helicopter 200 m above the ground sees a car directly ahead at angle of depression 15°. The helicopter then moves 100 m horizontally in the direction of the car. From its new position, what is the new angle of depression (1 d.p.)?

Reveal solution

Initial horizontal: $200/\tan 15° \approx 746.41$ m. New horizontal: $746.41 - 100 = 646.41$ m. New $\theta = \tan^{-1}(200/646.41) \approx 17.2°$.

Quick Review

Depression

From horizontal looking DOWN

= Elevation

Alternate angles — equal numbers

Adj = horizontal

At observer's eye

Opp = drop

Vertical distance

Mostly tan

Drop + horizontal

Hyp = slant

Line of sight downward

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