Some Applications of Trigonometry
Two-triangle problems
Solve height problems that need two right triangles.
Many questions give two angles, so there are two right triangles. The routine:
- Draw both triangles carefully as you read.
- Find the side they share (usually the height).
- Write one equation for each triangle, then solve the two together.
Example: shadows. A tower's shadow is 40 m longer when the Sun is at 30° than when it is at 60°. Call the height h and the short shadow x:
- at 60°: h = x tan 60° = x√3
- at 30°: h = (x + 40) tan 30° = (x + 40)/√3
- set them equal: 3x = x + 40, so x = 20 and h = 20√3 m
Angles of depression move inside the triangle. The horizontal at the top is parallel to the ground, so the angle of depression from above equals the angle of elevation from below (alternate angles).
Common layouts:
| Situation | What to find |
|---|---|
| a flagstaff or statue on top of a building | (height to the top) minus (height of the building) |
| two ships, or two points, one behind the other | the difference of the two ground distances |
| a bridge, or points on opposite sides | the sum of the two ground distances |
| a balloon, car or boat that moves | distance moved = the difference of the two distances; then speed = distance ÷ time |
Which way did it move? If the angle of elevation gets smaller, the object moved away. If it gets bigger, it came closer.
Read exactly what is asked. If they want the flagstaff, don't give the tower plus the flagstaff. And '6 seconds' is a time, not a distance: call the speed x, so the distance is 6x.
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