An angle of elevation is measured upward from an observer’s horizontal line to the line of sight. An angle of depression is measured downward from the observer’s horizontal line to the line of sight.
This is part of IB Mathematics: Applications and Interpretation SL 3.3: Angles of elevation and depression, which is common to both SL and HL.
| Feature | Angle of elevation | Angle of depression |
|---|---|---|
| Direction of sight | Upward | Downward |
| Starting line | Horizontal through the observer | Horizontal through the observer |
| Typical situation | Looking from the ground to the top of a building | Looking from a building down to a car |
| Position of angle | At the lower observer | At the upper observer |
Why the Two Angles Can Be Equal
Suppose a person at the top of a lighthouse looks down at a boat. Draw a horizontal line through the person and another through the boat. These horizontal lines are parallel, while the line of sight acts as a transversal.
Therefore, the angle of depression from the lighthouse equals the angle of elevation from the boat because they are alternate interior angles. This equality applies when both angles refer to the same line of sight and parallel horizontal lines.
For example, if the angle of depression to a boat is , the boat’s angle of elevation to the observer is also . If the horizontal distance is and the vertical separation is , then
so
A common misconception is measuring the angle from the vertical side. Both angles must be measured from a horizontal line, never from the vertical.
IB Exam Technique
In Paper 1 or Paper 2, first draw and label the horizontal, vertical, and line of sight. Keep your GDC in degree mode at SL, write the trigonometric equation before calculating, retain full precision, and give the final answer to three significant figures with units unless instructed otherwise.
