Apply trigonometric ratios to solve real problems involving heights, distances, and angles of elevation or depression.
The angle of elevation is measured upward from the horizontal to an object. Trigonometric ratios relate this angle to height and distance.
From a point 50 m from a tower's base, the angle of elevation to its top is 30°. Find the tower's height (tan30°≈0.577).
height=50×tan30°≈50×0.577≈28.87 m
A ladder makes a 60° angle with the ground and reaches 10 m up a wall. Find the ladder's length (sin60°≈0.866).
length=10/sin60°≈10/0.866≈11.55 m
The angle of depression is measured downward from the horizontal to an object below. It equals the angle of elevation from the object's viewpoint (alternate angles).
From the top of a 40 m cliff, the angle of depression to a boat is 45°. Find the horizontal distance to the boat.
Since tan45°=1, distance=height=40 m
Why does the angle of depression from A to B equal the angle of elevation from B to A?
Because they are alternate angles formed by a transversal (the line of sight) between two parallel horizontal lines
1. From a point 30 m from a building's base, the angle of elevation to its top is 45°. Find the height.
2. A kite string makes a 60° angle with the ground, and the string is 20 m long. Find the kite's height (sin60°≈0.866).
3. From a lighthouse 60 m tall, the angle of depression to a ship is 30°. Find the distance to the ship (tan30°≈0.577).
4. A ladder 15 m long leans against a wall making a 60° angle with the ground. How high up the wall does it reach (sin60°≈0.866)?
5. From a point 100 m from a tower, the angle of elevation is 30°. Find the tower's height (tan30°≈0.577).