Angle of Elevation

The angle of elevation is the angle formed when we look upward from our eye level to an object positioned above us. This angle is measured from the horizontal line of sight to the line joining the observer's eye to the object.

In a right triangle model:

• Horizontal ground = adjacent side
• Vertical height (object above eye level) = opposite side
• Line of sight = hypotenuse

Imagine you are standing on the ground looking at the top of a tree. Draw a horizontal line from your eye level. Now join your eye to the top of the tree with another line. The angle between the horizontal line and the line of sight is the angle of elevation.

This setup naturally forms a right-angled triangle, which is why trigonometry is used here.

Since the situation forms a right triangle, we use the trigonometric ratios:

• tan θ relates height and distance:

  \( \tan \theta = \dfrac{\text{height}}{\text{horizontal distance}} \)

• sin θ relates height and line of sight
• cos θ relates distance and line of sight

Most angle-of-elevation questions use tan θ because height and horizontal distance are usually known or required.

Example: You observe the top of a building at an angle of elevation of \(30^\circ\). If you are standing 20 m away from the building, find its height above your eye level.

This height is measured from your eye-level to the top of the building.