Apply trigonometric ratios to solve real-world problems involving angles of elevation and depression, including two-observation problems.
Basic problems use a single right triangle formed by the observer, the object, and the horizontal ground.
From a point 40 m from a tower, the angle of elevation is 30°. Find the height (tan30°≈0.577).
40×0.577≈23.09 m
From the top of a 50 m cliff, the angle of depression to a boat is 60°. Find the horizontal distance (tan60°≈1.732).
50/1.732≈28.87 m
More complex problems involve two angles of elevation from different distances, requiring two equations to be solved together.
From two points 20 m apart on the same side of a tower, the angles of elevation are 30° and 60°. Find the tower's height.
Using h/tan30°−h/tan60°=20; h(1.732−0.577)=20; h≈17.32 m
Why do two-position problems require setting up two separate equations?
Because each observation point gives a different angle and distance relationship, and both must be used together to solve for the unknown height and distance
1. From a point 60 m from a tower, the angle of elevation is 45°. Find the height.
2. From the top of a 30 m building, the angle of depression to a car is 30°. Find the distance (tan30°≈0.577).
3. A tower's angle of elevation from a point is 60°. If the point is 25 m from the base, find the height (tan60°≈1.732).
4. From two points 15 m apart, angles of elevation to a tower are 30° and 45°. Set up the equation for height h.
5. A flagpole casts a shadow of 12 m when the angle of elevation of the sun is 45°. Find the flagpole's height.