Right triangle trigonometry

The entirety of trigonometry is based on similarity and right triangles.

Contents

Trigonometric ratios

Hypotenuse is longest side in right triangle; opposite and adjacent are in relation to the angle in question
Hypotenuse, opposite and adjacent in relation to an acute angle in a right triangle

For an acute angle θ\theta in a right triangle, we define

sin⁡θ=oppositehypotenuse\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}
cos⁡θ=adjacenthypotenuse\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}
tan⁡θ=oppositeadjacent\tan\theta = \frac{\text{opposite}}{\text{adjacent}}

as ratios of side lengths in a right triangle.

For similar triangles, sin⁡θ\sin\theta, cos⁡θ\cos\theta, and tan⁡θ\tan\theta remain the same, despite changes in side lengths. In other words, the trigonometric ratios only depend on the angle, so they are also called trigonometric functions.

In right triangles, trig ratios are useful when you need to find a missing side.

Inverse trigonometric functions

Inverse trig functions sin⁡−1(x)\sin^{-1}(x), cos⁡−1(x)\cos^{-1}(x), tan⁡−1(x)\tan^{-1}(x) return the angle when you have the ratio xx. They are useful for missing angles.

Applications

definitions

The angle of elevation is the angle from horizontal up to some point of interest. The angle of depression is the angle from horizontal down to some point of interest.

strategies

  • Draw a diagram, label important points and given angles and sides (distances).
  • Light travels in a straight line.
  • Identify right angles.
  • Identify similar triangles.
  • Reuse variables when possible. For example if total distance is 1010, you can label part of it xx and the remaining 10−x10 - x.
  • Verify your calculator is in degrees mode, when working with angles in degrees.