AFCAT Numerical Ability · Heights and Distances

One Angle: Towers, Shadows, Ladders and Cliffs

One right triangle with one known side and one known angle: tangent links a height to a horizontal distance, sine and cosine link either of them to a slanting length.

All Heights and Distances notes

Why this matters

Most questions on this page are EASY, and they are where the chapter's marks are cheapest: a tower, a shadow, a ladder, a kite or a ship seen from a cliff. Marks are lost by using tangent when the known side is a slanting length, and by putting the angle of depression at the wrong corner of the triangle.

Concept 1 of 4: Height and horizontal distance: the tangent

The angle of elevation is the angle between the horizontal and the line from your eye up to the top of an object. The height and the horizontal distance are the two shorter sides of a right triangle, so tangent connects them: height over distance. A shadow is the same triangle, with the sun's elevation at the tip of the shadow.

Definition

$\tan\theta = \dfrac{\text{height}}{\text{distance}}$ , so height $= \text{distance} \times \tan\theta$ and distance $= \text{height} \times \cot\theta$ .
A shadow on level ground is the horizontal side; the sun's elevation is the angle at its far end.
When the angle is given as $\sin^{-1}$ or $\cos^{-1}$ of a fraction, draw the triangle for that fraction first and read off the tangent.
At 45° the height equals the distance; at 60° the height is $\sqrt3$ times the distance; at 30° it is $\tfrac{1}{\sqrt3}$ times.

Height from distance

h = d\tan\theta, \qquad d = h\cot\theta

Worked example

From a point 40 m from the foot of a tower, the angle of elevation of its top is 60°. Find the height of the tower.

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The same idea in a practice question:

Practice questionEasy

Example 1 · Heights and Distances · One Angle of Elevation or Depression

The angle of elevation of the top of the tower from a point on the ground is

\sin^{-1}(3/5)

. If the point of observation is 20 m away from the foot of the tower, what is the height of the tower?

Tangent uses the horizontal distance

Tangent links the height to the ground distance. If the length given is the line of sight or a rope, it is the hypotenuse and needs sine or cosine (next concept).

Height over distance, not the reverse

At 60°, a point 40 m away sees a tower

40\sqrt3

m tall. Writing

\tan 60^\circ = \tfrac{40}{h}

gives

\tfrac{40}{\sqrt3}

, the answer for 30°.

Drill 7 more on height and horizontal distance: the tangent

Concept 2 of 4: Ladders, ropes, kites and ramps: sine and cosine

A ladder, a kite string, a rope or a ramp is the slanting side, the hypotenuse. Its height is the length times the sine of its angle with the ground, and its foot is the length times the cosine. A tree or post that snaps forms the same triangle: the part still standing and the broken part that reaches the ground.

Definition

Height $= L\sin\theta$ , horizontal $= L\cos\theta$ , where $L$ is the slanting length and $\theta$ its angle with the ground.
Length from height: $L = \dfrac{h}{\sin\theta} = h\operatorname{cosec}\theta$ .
Broken post: the standing part is $d\tan\theta$ , the broken part is $\dfrac{d}{\cos\theta}$ , where $d$ is how far from the foot the top lands. The original height is their sum.
An angle with the wall is the complement of the angle with the ground.

A slanting length

h = L\sin\theta, \qquad x = L\cos\theta

Worked example

A ladder 12 m long rests against a wall, making 60° with the ground. How high up the wall does it reach, and how far is its foot from the wall?

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The same idea in a practice question:

Practice questionModerate

Example 2 · Heights and Distances · One Angle of Elevation or Depression

A telegraph post gets broken at a point against a storm and its top touches the ground at a distance 20 m from the base of the post making an angle

30^\circ

with the ground. What is the height of the post?

A length is the hypotenuse

The height a 12 m ladder reaches at 60° is

12\sin 60^\circ = 6\sqrt3

m. Using

12\tan 60^\circ = 12\sqrt3

treats the ladder as lying on the ground.

A broken tree is two pieces

The tree's height is the standing part plus the broken part. Giving the broken part alone, or the standing part alone, is an option set to catch you.

Drill 4 more on ladders, ropes, kites and ramps: sine and cosine

Concept 3 of 4: The angle of depression

The angle of depression is measured at the observer, from the horizontal line through the eye down to the object. The horizontal at the top and the ground are parallel, so this angle equals the angle of elevation of the observer as seen from the object. Put the angle at the object's corner and solve the same triangle as before.

Definition

Angle of depression from the top $=$ angle of elevation from the object (alternate angles between parallel lines).
The height of the observer is the vertical side; the distance to the object is the horizontal side.
Depression is measured from the horizontal. An angle of 20° with the vertical is a depression of 70°.

Depression equals elevation

\tan\theta = \dfrac{\text{height of observer}}{\text{horizontal distance}}

Worked example

From the top of a cliff 75 m high, a boat is seen at an angle of depression of 30°. How far is the boat from the foot of the cliff?

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The same idea in a practice question:

Practice questionModerate

Example 3 · Heights and Distances · One Angle of Elevation or Depression

From a cliff top, a boat is observed at

60°

angle of depression. The boat is

100

m horizontally from cliff base. What is the cliff height?

Measure from the horizontal

The angle of depression is between the horizontal and the line of sight. If a question gives the angle with the vertical, subtract it from 90° first.

Move the angle to the ground

Inside the triangle, the angle at the top is the complement of the depression. Using that corner with tangent gives the reciprocal answer. Put the depression at the object instead.

Drill 3 more on the angle of depression

Concept 4 of 4: Finding the angle of elevation

Sometimes the sides are known and the angle is asked. Take the rise and the run of the line of sight, divide, and look the tangent up. The rise is the height above the eye, so from the top of a shorter pole, or from a person's eye, it is the difference in heights.

Definition

$\tan\theta = \dfrac{\text{rise}}{\text{run}}$ . If the value is standard, name the angle; if not, the answer is $\tan^{-1}$ of the fraction.
From the top of one pole to the top of another, the rise is the difference of the heights and the run is the gap between them.
From a person's eye, subtract the eye height from the height of the object.
The same fraction gives the angle of depression from the higher top down to the lower one.

The angle from the sides

\theta = \tan^{-1}\dfrac{\text{rise}}{\text{run}}

Worked example

Two poles 8 m and 20 m tall stand 12 m apart. Find the angle of elevation of the top of the taller pole from the top of the shorter one.

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The same idea in a practice question:

Practice questionModerate

Example 4 · Heights and Distances · One Angle of Elevation or Depression

A person of height 2 m wants to get a fruit which is on a pole of height

(10/3)

m. If he stands at a distance of

(4/\sqrt{3})

m from the foot of the pole, then the angle at which he should throw the stone, so that it hits the fruit is

Use the difference of the heights

From the top of an 8 m pole to the top of a 20 m pole, the rise is 12 m. Using 20 m treats the observer as standing on the ground.

Not every angle is standard

\tan\theta = \tfrac27

, no table angle fits, and the right option reads

\tan^{-1}\tfrac27

. Do not round it to the nearest table angle.

Drill 3 more on finding the angle of elevation

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Height and horizontal distance: the tangent
Height from distance
$h = d\tan\theta, \qquad d = h\cot\theta$
Ladders, ropes, kites and ramps: sine and cosine
A slanting length
$h = L\sin\theta, \qquad x = L\cos\theta$
The angle of depression
Depression equals elevation
$\tan\theta = \dfrac{\text{height of observer}}{\text{horizontal distance}}$
Finding the angle of elevation
The angle from the sides
$\theta = \tan^{-1}\dfrac{\text{rise}}{\text{run}}$

Watch out for (8)

Tangent uses the horizontal distance
→ Height and horizontal distance: the tangent
Height over distance, not the reverse
→ Height and horizontal distance: the tangent
A length is the hypotenuse
→ Ladders, ropes, kites and ramps: sine and cosine
A broken tree is two pieces
→ Ladders, ropes, kites and ramps: sine and cosine
Measure from the horizontal
→ The angle of depression
Move the angle to the ground
→ The angle of depression
Use the difference of the heights
→ Finding the angle of elevation
Not every angle is standard
→ Finding the angle of elevation

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