Angles of elevation and depression help you use right triangles to measure heights and distances that are hard to reach directly. An angle of elevation is measured upward from a horizontal line of sight, such as looking from the ground to the top of a building. An angle of depression is measured downward from a horizontal line, such as looking from a tower to a person on the ground.
These angles are common in surveying, navigation, architecture, and physics problems involving lines of sight.
Key Facts
- Angle of elevation is measured up from a horizontal line to the line of sight.
- Angle of depression is measured down from a horizontal line to the line of sight.
- Horizontal lines are parallel, so the angle of depression from the top equals the angle of elevation from the ground when viewing the same two points.
- tan(theta) = opposite / adjacent is often used for height and distance problems.
- sin(theta) = opposite / hypotenuse and cos(theta) = adjacent / hypotenuse are useful when the line of sight length is given.
- If eye height matters, total object height = calculated vertical difference + observer eye height.
Vocabulary
- Angle of elevation
- The angle measured upward from a horizontal line to a line of sight.
- Angle of depression
- The angle measured downward from a horizontal line to a line of sight.
- Line of sight
- The straight path from an observer's eye to the object being viewed.
- Horizontal
- A level line that is parallel to the ground in a typical diagram.
- Right triangle
- A triangle with one 90 degree angle, often formed by height, horizontal distance, and line of sight.
Common Mistakes to Avoid
- Using the wrong reference line: angles of elevation and depression are measured from a horizontal line, not from the vertical side of the triangle.
- Choosing the wrong trig ratio: if the problem gives height and horizontal distance, tangent is usually the correct ratio because tan(theta) = opposite / adjacent.
- Forgetting eye height: when the observer's eyes are above the ground, the trig calculation gives the height above eye level, not always the total height of the object.
- Rounding too early: rounding intermediate values can noticeably change the final answer, so keep extra decimal places until the last step.
Practice Questions
- 1 A student stands 40 m from the base of a tower and measures an angle of elevation of 35 degrees to the top. If the student's eye height is 1.6 m, what is the total height of the tower?
- 2 From the top of a lighthouse, the angle of depression to a boat is 12 degrees. If the lighthouse is 55 m tall, how far is the boat horizontally from the base of the lighthouse?
- 3 A person on the ground looks up at a drone, and the drone camera looks down at the person. Explain why the angle of elevation from the person equals the angle of depression from the drone when the ground and drone's horizontal reference line are parallel.