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hscscience Maths Std · Y11
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Module 2 · L16 of 22 ~55 min MS-M2 · HIGH ⚡ +95 XP available

Angles of Elevation and Depression

Draw the horizontal reference line first, always. The angle opens between that horizontal and your line of sight, never from the vertical. Get the diagram right and the trigonometry is straightforward.

elev dep elevation = depression
Today's hook, You are on a cliff looking down at a boat. Your friend in the boat looks up at you. You both measure the angle from the horizontal to your line of sight. Do you get the same angle or different angles? Why?
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Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question
01
Think First, gut answer before the lesson
+5 XP warm-up

You are standing on top of a cliff looking down at a boat. Your friend is in the boat looking up at you. You both measure the angle between the horizontal and your line of sight to each other. Will you get the same angle or different angles? Why?

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02
The horizontal reference line, always draw it first
+5 XP to read

Every elevation and depression problem starts with a horizontal reference line drawn from the observer's eye. The angle opens between that horizontal and the line of sight, never from the vertical, never from the slope.

Angle of elevationmeasured upward from the horizontal to the line of sight when looking up at an object above.

Angle of depressionmeasured downward from the horizontal to the line of sight when looking down at an object below.

Key fact: The angle of elevation from A to B equals the angle of depression from B to A. Both equal $\theta$. This works because the two horizontal lines are parallel, the line of sight is a transversal, so the angles are alternate interior angles.

ANGLE OF ELEVATION horizontal θ object observer θ measured UP from horizontal ANGLE OF DEPRESSION horizontal θ object observer θ measured DOWN from horizontal
$\theta_{\text{elev}} = \theta_{\text{dep}}$, alternate interior angles between parallel horizontals
Most common ratio, tangent
In elevation/depression problems, O = vertical height and A = horizontal distance. Use $\tan\theta = O/A$. Only use $\sin$ or $\cos$ when the hypotenuse (line-of-sight distance) is given or required.
Equal angles, use them
When given a depression angle, redraw the triangle using the equal elevation angle at the base. The triangle is identical, same sides, same answer, but easier to visualise.
Two-observation problems
When two elevation angles give expressions for the same height $h$, set them equal. Solve for the unknown distance $d$, then back-substitute to find $h$.
03
What you'll master
Know

Key facts

  • Angle of elevation: measured upward from horizontal
  • Angle of depression: measured downward from horizontal
  • Both angles are always referenced from the horizontal
  • Elevation angle from A to B = depression angle from B to A
Understand

Concepts

  • Why the horizontal reference line must appear in every diagram
  • Why elevation and depression angles are equal (alternate angles)
  • How two-observation problems share a common height
Can do

Skills

  • Draw a correct labelled diagram from any word description
  • Solve elevation and depression problems using SOHCAHTOA
  • Apply the equal-angle relationship to simplify depression problems
  • Set up and solve two-observation simultaneous problems
04
Key terms
Angle of elevationThe angle measured upward from the horizontal to a line of sight, looking up at an object above you.
Angle of depressionThe angle measured downward from the horizontal to a line of sight, looking down at an object below you.
Horizontal reference lineA horizontal line drawn from the observer's eye level, the baseline from which elevation and depression angles are measured; must appear in every diagram.
Alternate interior anglesEqual angles formed on opposite sides of a transversal cutting two parallel lines, the geometric reason elevation and depression angles are equal.
05
Looking up, angle of elevation
core concept

The angle of elevation is measured from the horizontal upward to the line of sight. It is never measured from the ground if the ground is sloped, and never from the vertical.

Elements of a correct elevation diagram:

  • Observer at one end; horizontal reference line extending from observer
  • Line of sight goes upward from observer to the object
  • Angle of elevation sits between the horizontal and the line of sight
  • Right angle at the foot of the vertical height of the object
Common diagram error: Placing the right angle at the observer's position, or marking the angle from the vertical side. The right angle is at the base of the object's height, not at the observer.
$$h = d \cdot \tan\theta \qquad \text{where } d = \text{horizontal distance, } \theta = \text{elevation angle}$$

Angle of elevation: the angle measured upward from the horizontal to the line of sight to an object above. Angle of depression: measured downward from the horizontal to an object below. Both measured from the horizontal.

Pause, copy both definitions: angle of elevation = measured upward from the horizontal to the line of sight to an object above; angle of depression = measured downward from the horizontal to an object below, and note both are measured from the horizontal, not from the vertical into your book.

Did you get this? True or false: the angle of elevation is measured from the vertical to the line of sight.

Worked examples · 4 in a row, reveal as you go

PROBLEM 1 · BASIC ELEVATION

From a point 45 m from the base of a vertical building, the angle of elevation to the top is 38°. Find the height of the building, correct to 2 decimal places.

1
Horizontal distance $= 45$ m (A); height $= x$ (O); $\theta = 38°$. Use $\tan\theta = \dfrac{O}{A}$
No hypotenuse involved; O and A relate to the angle → tangent.
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PROBLEM 2 · ANGLE OF DEPRESSION

From the top of a cliff 80 m high, a boat is observed at sea with an angle of depression of 24°. Find the horizontal distance from the base of the cliff to the boat, correct to 2 decimal places.

1
Angle of depression $= 24°$ → angle of elevation from boat $= 24°$ (equal alternate angles)
Redraw from base: O = 80 m (cliff height), A = $x$ (horizontal distance), $\theta = 24°$.
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PROBLEM 3 · COMBINED HEIGHTS

A 12 m antenna sits on top of a 40 m building. An observer stands 75 m from the base of the building. Find the angle of elevation to the top of the antenna, to the nearest minute.

1
Total height $= 40 + 12 = 52$ m (O); horizontal distance $= 75$ m (A)
The observer is at ground level; the total height from ground to antenna top is the opposite side.
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Quick check: From the top of a 50 m building, the angle of depression to a car is 28°. What is the horizontal distance to the car?

09
Two angles to the same object

A single angle of elevation or depression problem produces one right-angled triangle with one trig equation to solve. When two angles to the same object are given, for example, two observers at different heights looking at the same point, two separate right-angled triangles share a common side. Write a trig equation for each triangle, express the shared side from each equation, then equate or substitute to eliminate the unknown.

When two angles of elevation to the same object are given from different points, you have two expressions for the same height. Setting them equal produces an equation solvable for the unknown distance.

Strategy:

  1. Let height $= h$, distance from nearer point to base $= d$
  2. Write $h = d \cdot \tan\theta_1$ (nearer point, larger angle)
  3. Write $h = (d + k) \cdot \tan\theta_2$ (further point, smaller angle, $k$ = separation)
  4. Set equal and solve for $d$
  5. Substitute back to find $h$
Check: The nearer point always has the larger angle of elevation. If your algebra gives a negative distance, you have labelled the points the wrong way around.

Two-angle problems use two separate right-angled triangles sharing a common side. Set up both triangles, label sides, write two trig equations, then solve simultaneously, often expressing the shared side in terms of the unknown angle or distance.

Pause, copy the two-triangle method: (1) draw and label both triangles clearly; (2) write a trig equation for each; (3) express the shared unknown side from each equation; (4) set the two expressions equal and solve algebraically into your book.

PROBLEM 4 · TWO-OBSERVATION

From point A, the angle of elevation to the top of a tower is 52°. From point B, 30 m further from the tower on the same side, the angle of elevation is 38°. Find the height of the tower, correct to 2 decimal places.

1
Let $d$ = distance from A to tower base; height $= h$.
From A: $h = d\tan 52°$   … (1)
From B: $h = (d+30)\tan 38°$   … (2)
Two triangles sharing the same height $h$; A is nearer (larger angle).
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Common errors · the 3 traps that cost marks

Trap 01
Omitting the horizontal reference line from the diagram
Without the horizontal reference line, it is impossible to correctly identify the angle of elevation or depression. Draw it first, before anything else, in every diagram, even rough sketches.
Trap 02
Measuring the angle from the vertical instead of the horizontal
Elevation and depression are always measured from the horizontal. If the angle is 30° from horizontal, it is not 30° from vertical (that would be the complement, 60°). Marking the angle wrong corrupts all subsequent working.
Trap 03
Using the wrong trig ratio in the two-observation setup
In two-observation problems, both triangles share the same height $h$ but have different horizontal distances. Write separate equations for each triangle and set them equal, do not average the angles or add the distances before writing the equations.

Fill the gap: The angle of depression from B down to A equals the angle of from A up to B, because the two horizontal lines are parallel, the angles formed are interior angles.

Quick-fire practice · 6 calculations

1

From a point 60 m from the base of a vertical tree, the angle of elevation to the top is 42°. Find the height of the tree to 2 d.p.

2

From the top of a 120 m cliff, the angle of depression to a boat is 31°. Find the horizontal distance from the cliff base to the boat, to 2 d.p.

3

An observer on the ground is 50 m from a building. The top is 35 m above the observer's eye level. Find the angle of elevation to the nearest minute.

4

A 15 m flagpole stands on top of a 25 m building. An observer is 80 m from the base. Find the angle of elevation to the top of the flagpole, to the nearest minute.

5

From point P, the angle of elevation to the top of a tower is 45°. From point Q, 20 m further from the tower than P, the angle of elevation is 30°. Find the height of the tower to 2 d.p.

6

From a lighthouse 65 m above sea level, a ship is observed with an angle of depression of 18°. Find the straight-line distance (line of sight) from the lighthouse to the ship, to 2 d.p.

Odd one out: Three of these statements are correct. Which one is wrong?

10
Revisit your thinking

You and your friend on the cliff both measure the angle from the horizontal to your mutual line of sight. The answer: you get the same angle. The angle of depression from the cliff top to the boat equals the angle of elevation from the boat to the cliff top, they are alternate interior angles between the two parallel horizontal lines cut by the line of sight.

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01
Multiple choice
+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidencethat tells the system what to drill next. Each retry pulls a fresh mix from the bank.

02
Short answer
ApplyBand 43 marks

SA 1. From a point on level ground, the angle of elevation to the top of a vertical cliff is 34°. The point is 220 m from the base of the cliff. (a) Draw a fully labelled diagram of this situation, including the horizontal reference line, angle, right angle, and all known measurements. (b) Find the height of the cliff correct to 2 decimal places. (3 marks)

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ApplyBand 43 marks

SA 2. A ship is sailing toward a 45 m lighthouse at sea level. From the ship, the angle of elevation to the top of the lighthouse is 12°. (a) Find the horizontal distance from the ship to the base of the lighthouse, to the nearest metre. (b) After the ship travels closer, the angle of elevation becomes 28°. How far has the ship travelled? Give your answer to the nearest metre. (3 marks)

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AnalyseBand 54 marks

SA 3. From point A on flat ground, the angle of elevation to the top of a vertical tower is 55°. From point B, which is 40 m further from the tower than A and on the same side, the angle of elevation is 35°. (a) Let the horizontal distance from A to the base of the tower be $d$ metres and the height be $h$ metres. Write two equations for $h$ in terms of $d$. (b) Find the value of $d$ correct to 2 decimal places. (c) Hence find the height of the tower correct to 2 decimal places. (4 marks)

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📖 Comprehensive answers (click to reveal)

Drill answers: 1: $h = 60\tan42° \approx \mathbf{54.05 \text{ m}}$  ·  2: $d = 120/\tan31° \approx \mathbf{199.71 \text{ m}}$  ·  3: $\tan^{-1}(35/50) \approx \mathbf{34°59'}$  ·  4: total $= 40$ m; $\tan\theta = 40/80$; $\approx \mathbf{26°34'}$  ·  5: $h = d = 20/(\sqrt{3}-1) \approx \mathbf{27.32 \text{ m}}$  ·  6: $H = 65/\sin18° \approx \mathbf{210.34 \text{ m}}$

SA 1 (3 marks): (a) Horizontal ground; 34° elevation angle; horizontal reference from observer; right angle at cliff base; 220 m horizontal; height $x$ [1]. (b) $\tan34° = x/220$; $x = 220\tan34° = 220 \times 0.67450\ldots \approx \mathbf{148.39 \text{ m}}$ [2].

SA 2 (3 marks): (a) $d_1 = 45/\tan12° \approx 45/0.21256 \approx 211.73 \approx \mathbf{212 \text{ m}}$ [2]. (b) $d_2 = 45/\tan28° \approx 45/0.53171 \approx 84.64 \approx 85$ m; distance $= 212 - 85 = \mathbf{127 \text{ m}}$ [1].

SA 3 (4 marks): (a) $h = d\tan55°$ and $h = (d+40)\tan35°$ [1]. (b) $d\tan55° = (d+40)\tan35°$; $d(1.42814 - 0.70021) = 40 \times 0.70021$; $d \times 0.72793 = 28.008$; $d \approx \mathbf{38.48 \text{ m}}$ [2]. (c) $h = 38.48 \times \tan55° = 38.48 \times 1.42814 \approx \mathbf{54.95 \text{ m}}$ [1].

01
Boss battle · The Surveyor
earn bronze · silver · gold

Five timed questions on elevation and depression. Beat the boss to bank a tier, gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.

⚔ Enter the arena
02
Science Jump · platform challenge

Climb platforms by answering elevation and depression questions. Pool: lesson 16.

Mark lesson as complete

Tick when you've finished the practice and review.

Module overview · Maths Standard Y11