Applications of trigonometric ratios

Use ratios suited to given angles and lengths to find heights, distances, and areas.

Meaning

Trigonometric ratios connect angles and lengths when a right triangle can model a situation. Identify which sides form the measured angle and choose the appropriate ratio.

Example

On level ground, a tree is horizontally 10 m away and its top has angle of elevation 45 degrees from eye level. Its height above eye level is m.Relative to the 45-degree angle, the known 10 m distance is adjacent and the unknown height above eye level is opposite. Tangent connects exactly these two sides: h/10=tan45°, so h=10tan45°.Height above the eyes differs from total height above ground. Omitting eye height or ground slope can change a real-world answer.

Practice for this concept

Scope and references

Topics

Finding heights and distances, triangle areas

Learning goal

Use ratios suited to given angles and lengths to find heights, distances, and areas.

Helpful background

Check side positions relative to the angle and the ratios.