Apply trigonometric ratios to solve real problems involving heights and distances, using angles of elevation and depression.
The angle of elevation is measured upward from horizontal. Trigonometric ratios connect this angle to height and horizontal distance.
From a point 30m from a tower, the angle of elevation is 30°. Find the tower's height (tan30°≈0.577).
30×0.577≈17.32m
A ladder makes a 60° angle with the ground, reaching 15m up a wall. Find the ladder's length (sin60°≈0.866).
15/0.866≈17.32m
The angle of depression is measured downward from horizontal, and equals the angle of elevation from the object's viewpoint due to alternate angles.
From a 50m cliff, the angle of depression to a boat is 30°. Find the distance (tan30°≈0.577).
50/0.577≈86.66m
Two buildings are 20m apart. From the top of one (40m tall), the angle of depression to the base of the other is 45°. Find the height difference.
Since tan45°=1, height difference=distance=20m; other building's height=40-20=20m
1. From a point 40m from a tower, the angle of elevation is 45°. Find the height.
2. A kite string 25m long makes a 60° angle with the ground. Find the kite's height (sin60°≈0.866).
3. From a lighthouse 45m tall, the angle of depression to a ship is 30°. Find the distance (tan30°≈0.577).
4. A tower's angle of elevation from a point 50m away is 60°. Find its height (tan60°≈1.732).
5. From two points 25m apart, angles of elevation to a tower are 30° and 60°. Set up the height equation.