Angles of elevation and depression

Two crucial vocabulary items power this chapter.

Definitions

The angle of elevation of a point $P$ from an observer at $O$ is the angle that the line of sight $OP$ makes with the horizontal through $O$, measured upward. It is used when the object is above the observer.

The angle of depression of a point $P$ from an observer at $O$ is the angle that the line of sight $OP$ makes with the horizontal through $O$, measured downward. It is used when the object is below the observer.

Key facts

• The angle of elevation of $P$ from $O$ equals the angle of depression of $O$ from $P$. They are alternate interior angles between parallel horizontals.
• All angles are measured from the horizontal, never from the vertical.
• The horizontal is the level eye-line of the observer.
• The object's height is measured from the base level (typically the ground).

Setting up a problem

A reliable five-step recipe:

1. Read the problem twice. Identify the observer and the object.
2. Draw a diagram. Put the observer at the corner of a right triangle, with the horizontal leg on the ground and the vertical leg as the height.
3. Mark all given lengths and angles. Use letters for the unknowns.
4. Pick a trig ratio. Usually $\tan$, because the legs are horizontal and vertical and we know one of them.
5. Solve. Use the standard-angle table if the angle is $30^\circ, 45^\circ, 60^\circ$. Verify the answer makes physical sense.

Worked examples

Example 1. The angle of elevation of the top of a tower from a point on the ground, $30$ m away, is $30^\circ$. Find the height of the tower.

Let height $= h$. Then $\tan 30^\circ = h/30 \Rightarrow h = 30 \cdot 1/\sqrt{3} = 10 \sqrt{3} \approx 17.3$ m.

Example 2. From the top of a building $20$ m high, the angle of depression of a car on the road is $30^\circ$. Find the distance of the car from the building.

Let the distance $= d$. The angle of depression is $30^\circ$, so the angle between the line of sight and the horizontal at the top is $30^\circ$.

$\tan 30^\circ = 20/d \Rightarrow d = 20/\tan 30^\circ = 20 \sqrt{3} \approx 34.6$ m.

Example 3. A ladder leans against a wall at $60^\circ$ to the ground. If the foot of the ladder is $4$ m from the wall, how long is the ladder?

The ladder is the hypotenuse, the horizontal is $4$ m. $\cos 60^\circ = 4/L \Rightarrow L = 4/(1/2) = 8$ m.

Example 4. From a point on the ground, the angle of elevation of the top of a flagstaff is $60^\circ$. The flagstaff is $20$ m tall. Find the distance from the observer to the flagstaff.

$\tan 60^\circ = 20/d \Rightarrow d = 20/\sqrt{3} = (20\sqrt{3})/3$ m.

Example 5. An observer $1.5$ m tall is $30$ m away from a tower. The angle of elevation of the top of the tower from the eye of the observer is $30^\circ$. Find the height of the tower.

Eye-level is at $1.5$ m. Effective elevation: height from eye-level. $\tan 30^\circ = h'/30 \Rightarrow h' = 30/\sqrt{3} = 10\sqrt{3}$ m.

Total tower height $= h' + 1.5 = 10\sqrt{3} + 1.5 \approx 18.8$ m.

Try it yourself

1. The angle of elevation of a tower from $100$ m is $45^\circ$. Height?
2. The angle of depression of a boat from a $50$-m cliff is $30^\circ$. Find the boat's distance from the foot of the cliff.
3. A kite at height $60$ m is attached to a string of length $120$ m. What is the angle of inclination of the string?
4. A person on a $20$-m hill sees a tree at angle of depression $45^\circ$. Distance of tree?
5. A pole casts a shadow $\sqrt{3}$ times its own length. Find the sun's angle of elevation.
6. From the top of a $30$-m building, the angle of depression of a point on the ground is $45^\circ$. Find the distance of the point from the building.
7. The angle of elevation of the top of a tower from two points $a$ and $b$ from the foot of the tower in the same straight line is $30^\circ$ and $60^\circ$. (Set up only.)
8. If $\tan\theta = 5/12$ for the elevation of a $50$-m tower, find the distance from the observer.
9. A bridge $25$ m above a river makes an angle of depression of $45^\circ$ with a boat. Find the horizontal distance from the bridge to the boat.
10. An observer at sea level sees a lighthouse at an angle of elevation of $60^\circ$. The lighthouse is $30$ m tall. Distance?

Pitfalls / Insight

• The angle is measured from the horizontal, not the vertical.
• Eye-level vs. ground level , include the observer's height if asked.
• Diagram first. Algebra without a diagram is the leading cause of mistakes.

Insight. Heights and distances are just a careful choice of right triangle. Once it's drawn, the rest is one trig ratio away.