20.3 Angles of elevation and depression — Learn

Prerequisites: trigonometric ratios and drawing right-angled triangles.

Objectives: model an angle of elevation or depression and solve for a distance or angle.

An angle of elevation is measured upwards from a horizontal line. An angle of depression is measured downwards from a horizontal line. The horizontal lines are parallel, so the angle of depression equals the matching angle inside the triangle.

Example 1. A person stands $20$ m from a tower. The angle of elevation to its top is $35^\circ$. Find the height above the person's eye level.

Draw a right-angled triangle. The height is opposite and $20$ m is adjacent.

Example 2. A boat is $50$ m below a cliff top. The angle of depression is $25^\circ$. Find the horizontal distance $d$.

The angle inside the boat triangle is also $25^\circ$ because the horizontal lines are parallel.

Watch out. The angle of depression starts at the observer's horizontal line. Copying it into the triangle uses parallel lines; check the matching angle before choosing a ratio.

Progress check

Sketch the garden and its diagonal. Mark the right angle and explain which angle would be measured from a garden edge.

You should now be able to

• draw and label a right-angled triangle for an elevation or depression problem;
• select a trigonometric ratio to find a distance or angle;
• state whether a calculated height is above eye level or ground level.