Some Applications of Trigonometry

Find heights and distances you cannot measure directly, using angles of elevation and depression of 30°, 45° and 60°.

Key ideas

Draw the figure first: the object is vertical, the ground horizontal, so there is a right angle at the foot.
The angle of elevation is measured up from the horizontal; the angle of depression is measured down from the horizontal.
The angle of elevation of A from B equals the angle of depression of B from A (alternate angles).
tan θ links the height to the distance along the ground, which is what most problems need.
With two angles from two points, write the height twice (once for each angle) and solve for the unknown distance.
Keep answers exact (like 10√3 m) unless the question asks for a decimal.

Remember

Height from distance	height = distance × tan θ
Distance from height	distance = height ÷ tan θ
Values	tan 30° = 1/√3, tan 45° = 1, tan 60° = √3
Values	sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2

Watch out

Measuring the angle of depression from the vertical.
Using sin when the side you know is along the ground (use tan).
Forgetting to add the observer's height or the height of a building the observer stands on.

Worked example

From the top of a tower, a man sees a car moving towards the foot of the tower at a steady speed. The angle of depression of the car changes from 30° to 60° in 16 minutes. How much longer will the car take to reach the tower?

Working: with height h, the distances are h√3 and h/√3; the car covers 2h/√3 in 16 minutes, and h/√3 is half of that

Answer: 8 minutes