Lines of Sight, Elevation and Depression: Quick Revision

A brief note on Lines of Sight, Elevation and Depression. Read it before a test. Try each question at the end before you open its answer.

The words

• The line of sight is the straight line from the eye to the object.
• The horizontal level is the flat line through the eye.
• The angle of elevation is measured from the horizontal up to the line of sight.
• The angle of depression is measured from the horizontal down to the line of sight.
• For one line of sight, the two angles are equal. They are alternate angles between parallel horizontals.

The method

1. Draw the ground and the vertical object. Mark the right angle.
2. Draw the line of sight and mark the given angle from the horizontal.
3. Write the known lengths and a letter for the unknown one.
4. Name the sides from the marked angle, then choose the ratio.

Which ratio

• Height and ground distance: $\tan \theta = \dfrac{h}{d}$.
• Height and slanting length (ladder, string, wire): $\sin \theta = \dfrac{h}{L}$.
• Ground distance and slanting length: $\cos \theta = \dfrac{d}{L}$.

The values to know

• $\tan 30° = \dfrac{1}{\sqrt{3}}$, $\tan 45° = 1$ and $\tan 60° = \sqrt{3}$.
• $\sin 30° = \cos 60° = \dfrac{1}{2}$ and $\sin 60° = \cos 30° = \dfrac{\sqrt{3}}{2}$.
• $\sin 45° = \cos 45° = \dfrac{1}{\sqrt{2}}$.

A worked example

A tower is seen from $24\ \text{m}$ away at an angle of elevation of $30°$. Find its height, then check the angle.

1. Ratio. The height is opposite and the distance is adjacent, so $\tan 30° = \dfrac{h}{24}$.
2. Height. $h = \dfrac{24}{\sqrt{3}} = 8\sqrt{3} \approx 13.86\ \text{m}$.
3. Check. $\tan^{-1}\left(\dfrac{13.86}{24}\right) \approx 30°$.

Practice with new numbers

A program to try

The program finds the height from the distance and the angle, and then finds the angle back from the height and the distance.

Mistakes to avoid

• Measuring the angle from the vertical. Elevation and depression start at the horizontal.
• Multiplying instead of dividing. A height is $d \tan \theta$. A distance is $\dfrac{h}{\tan \theta}$.
• Forgetting the eye height. Add it at the end when the eye is above the ground.
• Mixing $\sin$ and $\cos$ for a ladder. The height reached uses $\sin$ and the foot distance uses $\cos$.

Check yourself

1. From the top of a hill, the angle of depression of a hut is $20°$. What is the angle of elevation of the hilltop seen from the hut?

2. A pole's shadow is $\sqrt{3}$ times its height. What is the sun's altitude?

3. A $10\ \text{m}$ ladder makes $30°$ with the ground. How high does it reach?

4. Find the distance of a boat from the foot of a $15\ \text{m}$ lighthouse when the angle of depression is $45°$.