Solving Right Triangles Using Trig

A right triangle is a triangle that has one angle measuring 90 degrees.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b): $$c^2 = a^2 + b^2$$ .

The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse: $$\text{sine}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$ .

The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse: $$\text{cosine}(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ .

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side: $$\text{tangent}(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ .

To find the height (h) of a triangle, use the sine function: $$h = \text{hypotenuse} \times \text{sine}(\theta)$$ .

To find the hypotenuse (c), use the sine or cosine function: $$c = \frac{\text{opposite}}{\text{sine}(\theta)}$$ or $$c = \frac{\text{adjacent}}{\text{cosine}(\theta)}$$ .

The angle of elevation is the angle formed by the line of sight when looking up from a horizontal line to an object above.

The angle of depression is the angle formed by the line of sight when looking down from a horizontal line to an object below.

Use trigonometric ratios: For example, if you know the angle of elevation and the height, use $$\text{distance} = \frac{\text{height}}{\text{tan}(\theta)}$$ .