Angles of Elevation & Depression

An angle of elevation is measured looking up: it opens from a horizontal line up to your line of sight. An angle of depression is measured looking down: it opens from a horizontal line down to your line of sight. Both start at the horizontal — never at a wall, a cliff face, or the ground under a tower.

These problems are just right-triangle trig wearing a word-problem costume. The height involved is one leg, the horizontal ground distance is the other leg, the line of sight is the hypotenuse, and one trig equation finds whatever is missing.

The two angles are congruent

Picture a person at the top of a lighthouse looking down at a boat. The horizontal line at the top of the lighthouse and the horizontal water line at the bottom are parallel, and the line of sight cuts across both. That makes the angle of depression at the top and the angle of elevation at the boat alternate interior angles — so they are equal.

This is the single most useful fact in the lesson. When a problem gives you an angle of depression, you can drop that same angle measure inside the triangle at the ground and work from there. The figure below shows why: the two horizontal lines are parallel, the line of sight is a transversal, and the depression angle at the top and the elevation angle at the bottom are congruent alternate interior angles.

Set up the right triangle

Draw the situation: a vertical height, a horizontal ground distance, and the slanted line of sight connecting them. The right angle sits where the vertical meets the horizontal.

Then label from the angle inside the triangle. The height is usually opposite the angle and the ground distance is usually adjacent, which is why tangent solves most elevation and depression problems: $\tan(\text{angle}) = \dfrac{\text{height}}{\text{ground distance}}$ . If the problem involves the line of sight itself — a wire, a string, a ramp surface — that is the hypotenuse, and you will need sine or cosine instead.

Where students go wrong

The most common error is measuring the depression angle from the vertical — down the cliff face — instead of from the horizontal. The angle of depression always opens between the horizontal sight line and the downward line of sight. If you measure from the vertical, you get the complement of the correct angle, and everything after that is wrong.

The second error is using tangent when the problem gives the line-of-sight length. A kite string, a guy wire, or a ladder is a hypotenuse, and tangent never uses the hypotenuse — pair the hypotenuse with the height using sine, or with the ground distance using cosine.

Worked examples

Example 1: angle of elevation, find the height

From a spot on the ground $40$ ft from the base of a building, the angle of elevation to the roof is $62^\circ$ . How tall is the building, to the nearest tenth?

The height is opposite the angle; the

40

-ft distance is adjacent — tangent

\tan 62^\circ = \dfrac{x}{40}

Multiply both sides by

40

x = 40\tan 62^\circ

Evaluate

x \approx 75.2

Answer: About $75.2$ ft

Example 2: angle of depression, find the distance

From the top of a $120$ -ft lighthouse, the angle of depression to a boat is $31^\circ$ . How far is the boat from the base of the lighthouse, to the nearest tenth?

The depression angle equals the elevation angle at the boat

31^\circ

The

120

-ft height is opposite that angle; the distance is adjacent

\tan 31^\circ = \dfrac{120}{x}

The unknown is on the bottom, so divide

x = \dfrac{120}{\tan 31^\circ}

Evaluate

x \approx 199.7

Answer: About $199.7$ ft

Example 3: the line of sight is the hypotenuse

A kite string is $85$ ft long and makes a $42^\circ$ angle of elevation with the level ground. How high is the kite, to the nearest tenth?

The string is the hypotenuse and the height is opposite the angle — sine, not tangent

\sin 42^\circ = \dfrac{x}{85}

Multiply both sides by

85

x = 85\sin 42^\circ

Evaluate

x \approx 56.9

Answer: About $56.9$ ft

Try one yourself

Common questions

Are the angle of elevation and the angle of depression always equal?

Yes, for the same two points. The two horizontal lines are parallel and the line of sight is a transversal, so the angles are alternate interior angles — congruent. The elevation angle sits at the lower point and the depression angle sits at the upper point.

Why is tangent used so often in these problems?

Because the two natural measurements — a height and a ground distance — are the two legs of the right triangle, and tangent is the ratio that connects the legs. When the problem instead gives the line-of-sight length, that is the hypotenuse and you switch to sine or cosine.

Is the angle of depression measured from the vertical or the horizontal?

Always from the horizontal. An observer looking down starts with a level sight line and rotates it down to the target — that rotation is the angle of depression. Measuring from a vertical wall or cliff gives the complement, which is a different angle.

What if the problem asks for the angle instead of a side?

Build the same triangle, write the ratio from the two known sides, and apply the matching inverse trig function — for example, an observer $50$ ft away from a $30$ -ft tree sees an elevation angle of $\tan^{-1}\left(\dfrac{30}{50}\right) \approx 31^\circ$ .

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Angles of Elevation & Depression

The two angles are congruent

Set up the right triangle

Where students go wrong

Worked examples

Example 1: angle of elevation, find the height

Example 2: angle of depression, find the distance

Example 3: the line of sight is the hypotenuse

Try one yourself

Common questions

Keep learning

Solving for Missing Sides with Trig

Inverse Trig: Finding Missing Angles

The Pythagorean Theorem & Its Converse