Year 11 General Applications Of Trigonometry

Angles Of Elevation And Depression

Q: What is the angle of elevation?

The angle of elevation is the angle measured upwards from the horizontal sight line to a point above the observer. It's always measured from the horizontal — never from the vertical.

Q: What is the angle of depression?

The angle of depression is the angle measured downwards from the horizontal sight line to a point below the observer. The observer is looking down, and again the angle is measured from the horizontal.

Q: Is the angle of elevation equal to the angle of depression?

Yes, between the same two points. If you stand at A and look up at B with an angle of elevation theta, then someone at B looking down at A sees an angle of depression of the same size theta. They are alternate angles between parallel horizontal sight lines. This is called the complement rule.

Q: How do you find the angle of elevation when you know the height and distance?

Use the inverse tangent function: theta equals tan inverse of height divided by horizontal distance. For example, if a tower is 57.6 m tall and you're standing 45 m away, the angle of elevation to the top is tan inverse of 57.6 divided by 45, approximately 52 degrees.

Q: Do I need to add my eye height to the answer?

Only if the question asks for the total height of the object. The trig calculation gives the height of the object above the observer's eye, so if the observer is 1.7 m tall, you add 1.7 m to get the full height from the ground.

Q: What's the difference between angle of elevation and bearing?

An angle of elevation is measured vertically from the horizontal up to an object. A bearing is measured horizontally from north (or another reference direction). They describe completely different planes — elevation is up/down, bearing is left/right.

12 practice questions 2 video lessons Theory + worked examples

Theory

The angle of elevation is measured upwards from the horizontal; the angle of depression is measured downwards. Both form a right triangle with the vertical height and horizontal distance, so every problem solves with SOH-CAH-TOA.

The angle of elevation is the angle measured upwards from the horizontal, looking up to an object above the observer.

The angle of depression is the angle measured downwards from the horizontal, looking down to an object below the observer.

Looking up from the horizontal

Looking down from the horizontal

Key formulas

For a right triangle with angle $θ$ measured from the horizontal:

\tan θ = \frac{opposite}{adjacent} = \frac{vertical height}{horizontal distance}

\tan θ = \frac{opposite}{adjacent} = \frac{vertical height}{horizontal distance}

\sin θ = \frac{opposite}{hypotenuse}

\sin θ = \frac{opposite}{hypotenuse}

\cos θ = \frac{adjacent}{hypotenuse}

\cos θ = \frac{adjacent}{hypotenuse}

          The complement rule: If you stand at point A and look up at point B with an angle of elevation θ, then someone at B looking back down at A sees an angle of depression of the same size θ. They are alternate angles between two parallel horizontal sight lines.
        

How to solve any elevation or depression problem

Sketch the situation. Draw the horizontal sight line, then the line to the object, and mark the elevation or depression angle.
Identify the right triangle: vertical height, horizontal distance, and the slant line of sight (hypotenuse).
Apply SOH-CAH-TOA to find the unknown side or angle.

Example 1 — Elevation: find the height

A radio tower stands on flat ground. From a point

45

m from its base, the angle of elevation to the top is

52^{\circ}

. Find the height of the tower (1 dp).

Solution

Sketch the right triangle. The height $h$ is opposite the $52^{\circ}$ angle, the $45$ m is adjacent:

Use $\tan$ :

$\tan 52^{\circ}$	$=$	$\frac{h}{45}$
$h$	$=$	$45 \tan 52^{\circ}$
$h$	$\approx$	$57.6 m$

\tan 52 ° = \frac{h}{45}, h = 45 \tan 52 ° \approx 57.6 m

Example 2 — Depression: find the horizontal distance

A drone hovers at

80

m altitude. The angle of depression to a car on the ground is

35^{\circ}

. Find the horizontal distance from the point directly below the drone to the car (1 dp).

Solution

Sketch the triangle. By the complement rule the angle at the car looking up at the drone is also $35^{\circ}$ . The $80$ m is opposite, $d$ is adjacent:

$\tan 35^{\circ}$	$=$	$\frac{80}{d}$
$d$	$=$	$\frac{80}{\tan 35^{\circ}}$
$d$	$\approx$	$114.3 m$

\tan 35 ° = \frac{80}{d}, d = \frac{80}{\tan 35 °} \approx 114.3 m

Example 3 — Eye-level adjustment

A person

1.7

m tall stands

30

m from a tree. The angle of elevation from their eye to the top of the tree is

42^{\circ}

. Find the total height of the tree (1 dp).

Solution

Sketch the triangle. Watch out — the triangle starts at eye height, not ground level. We first find $t$ , the height of the tree above eye level:

$\tan 42^{\circ}$	$=$	$\frac{t}{30}$
$t$	$=$	$30 \tan 42^{\circ}$
$t$	$\approx$	$27.0 m$

\tan 42 ° = \frac{t}{30}, t = 30 \tan 42 ° \approx 27.0 m

Then add the observer's eye height to get the total tree height:

total height = 27.0 + 1.7 = 28.7 m

Example 4 — Find the angle

A man

175

cm tall casts a shadow

190

cm long. Find the angle of elevation of the sun (nearest degree).

Solution

Sketch the triangle. The man's height is opposite the angle, the shadow length is adjacent:

Use $\tan$ :

$\tan θ$	$=$	$\frac{175}{190}$
$θ$	$=$	$\tan^{- 1} (\frac{175}{190})$
$θ$	$\approx$	$43^{\circ}$

\tan θ = \frac{175}{190}, θ = \tan^{- 1} (\frac{175}{190}) \approx 43 °

Common pitfalls

Always from the horizontal, never the vertical. If a problem gives an angle from the vertical (e.g. "

28^{\circ}

from the vertical"), subtract from

90^{\circ}

first.

Eye-level matters. If the observer's eye is not at ground level (e.g. a person

1.7

m tall looking up at a tree), the trig calculation gives the height of the object above eye level. Add the eye height for the total.

Two observations, two triangles. If the same object is viewed from two positions, you usually get two right triangles sharing one side (the height). Set up two equations and solve simultaneously.

Frequently asked questions

What is the angle of elevation?

The angle of elevation is the angle measured upwards from the horizontal sight line to a point above the observer. It's always measured from the horizontal — never from the vertical.

What is the angle of depression?

The angle of depression is the angle measured downwards from the horizontal sight line to a point below the observer. The observer is looking down, and again the angle is measured from the horizontal.

Is the angle of elevation equal to the angle of depression?

Yes — between the same two points. If you stand at $A$ and look up at $B$ with an angle of elevation $θ$ , then someone at $B$ looking down at $A$ sees an angle of depression of the same size $θ$ . They are alternate angles between parallel horizontal sight lines. This is called the complement rule.

How do you find the angle of elevation when you know the height and distance?

Use the inverse tangent function: $θ = \tan^{- 1} (\frac{height}{horizontal distance})$ . For example, if a tower is $57.6$ m tall and you're standing $45$ m away, the angle of elevation to the top is $\tan^{- 1} (57.6 / 45) \approx 52^{\circ}$ .

Do I need to add my eye height to the answer?

Only if the question asks for the total height of the object. The trig calculation gives the height of the object above the observer's eye, so if the observer is $1.7$ m tall, you add $1.7$ m to get the full height from the ground.

What's the difference between angle of elevation and bearing?

An angle of elevation is measured vertically from the horizontal up to an object. A bearing is measured horizontally from north (or another reference direction). They describe completely different planes — elevation is up/down, bearing is left/right.