Finding a distance

A distance from a height

Sometimes the height is known and the distance is wanted: how far is a ship from a lighthouse, or a car from a bridge?

The ground distance is adjacent to the angle and the height is opposite. From $\tan \theta = \dfrac{h}{d}$ we get

Notice that you divide by the tangent now. A larger angle makes the distance smaller. That matches the picture: if you are close to a tall tower, you must look steeply up.

A worked example. From the top of a lighthouse $36\ \text{m}$ high, the angle of depression of a boat is $60°$. How far is the boat from the foot of the lighthouse?

1. Use the equal angle. The angle of elevation of the lamp, seen from the boat, is $60°$.
2. Choose the ratio. The height is opposite and the distance is adjacent, so use $\tan$.
3. Write the relation. $\tan 60° = \dfrac{36}{d}$.
4. Solve. $d = \dfrac{36}{\sqrt{3}} = 12\sqrt{3} \approx 20.78\ \text{m}$.

A ladder against a wall

A ladder is a slanting length, so it works like the kite string. The ladder is the hypotenuse. The wall is vertical, so the height it reaches is opposite the angle at the ground, and the distance of the foot from the wall is adjacent.

Check the last readout. The height squared plus the foot distance squared always equals the ladder length squared. This is Pythagoras, and it is a good way to test any ladder answer.

A worked example. A ladder $12\ \text{m}$ long leans against a wall and makes $60°$ with the ground. How far is its foot from the wall, and how high does it reach?

1. The foot. $12 \cos 60° = 12 \times \dfrac{1}{2} = 6\ \text{m}$.
2. The height. $12 \sin 60° = 12 \times \dfrac{\sqrt{3}}{2} = 6\sqrt{3} \approx 10.39\ \text{m}$.
3. Check. $6^2 + 10.39^2 \approx 144 = 12^2$.

Shadows and the sun

The sun is very far away, so its rays reach the ground as parallel lines. A pole, its shadow and the ray from the top of the pole to the tip of the shadow make a right triangle. The angle at the tip of the shadow is the angle of elevation of the sun, also called the sun's altitude.

A low sun gives a long shadow. At noon in summer the shadow is short.

A worked example. A pole $10\ \text{m}$ high casts a shadow on level ground when the sun's altitude is $30°$. How long is the shadow?

1. Choose the ratio. The height is opposite and the shadow is adjacent, so use $\tan$.
2. Write the relation. $\tan 30° = \dfrac{10}{s}$.
3. Solve. $s = \dfrac{10}{\tan 30°} = 10\sqrt{3} \approx 17.32\ \text{m}$.

Finding the angle itself

So far the angle was given. Now suppose we know two lengths and want the angle. Find the ratio first. Then ask which angle has that ratio.

A worked example. A pole $7\ \text{m}$ high casts a shadow $4\ \text{m}$ long. Find the sun's altitude.

1. Find the ratio. $\tan \theta = \dfrac{7}{4} = 1.75$.
2. Go back to the angle. The inverse tangent, written $\tan^{-1}$, gives the angle whose tangent is $1.75$.
3. Use a calculator or Python. $\theta = \tan^{-1}(1.75) \approx 60.3°$.

In board papers the ratio is usually one you know, such as $\sqrt{3}$ or $\dfrac{1}{\sqrt{3}}$ or $1$. Then the angle is $60°$, $30°$ or $45°$ straight away.

Practise with new numbers

Try it in Python

This program finds the height reached and the foot distance for a $10\ \text{m}$ ladder at three angles. It also checks each answer with Pythagoras.

Common mistakes

• Multiplying when you should divide. The distance is the height divided by $\tan \theta$.
• Mixing up $\sin$ and $\cos$ for a ladder. The height reached uses $\sin$ and the foot uses $\cos$, when the angle is with the ground.
• Taking the shadow as the slanting length. The shadow lies on the ground, so it is the adjacent side.
• Giving $\tan \theta$ as the angle. After finding the ratio, you still have to find the angle that has it.

Remember these

• Ground distance: $d = \dfrac{h}{\tan \theta}$.
• Ladder at angle $\theta$ with the ground: height $L \sin \theta$ and foot $L \cos \theta$.
• Shadow: $s = \dfrac{h}{\tan \theta}$, where $\theta$ is the sun's altitude. A low sun means a long shadow.
• To find an angle, find its ratio first and then use the inverse function or a known value.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. From the top of a tower $25\ \text{m}$ high, the angle of depression of a car is $45°$. How far is the car from the foot of the tower?

2. A ladder $8\ \text{m}$ long makes $60°$ with the ground. How far is its foot from the wall?

3. The sun's altitude falls from $60°$ to $30°$. What happens to the shadow of a pole?

4. A pole and its shadow are both $13\ \text{m}$ long. What is the sun's altitude?

5. Is it possible for a ladder $5\ \text{m}$ long to reach $6\ \text{m}$ up a wall?