# Trigonometric Functions

Geometric definitions of trigonometric functions for real values.

## Right triangle

For angles $$0 \lt \alpha \lt \frac{\pi}{2}$$, the trigonometric functions can be defined as ratios of side lengths in a right triangle.

\begin{align*} \sin \alpha &= \frac{\text{opposite}}{\text{hypotenuse}}\\ \cos \alpha &= \frac{\text{adjacent}}{\text{hypotenuse}}\\ \tan \alpha &= \frac{\text{opposite}}{\text{adjacent}} \end{align*}

Another three are obtained via reciprocals: The cosecant ($$\csc$$) from the sine, the secant ($$\sec$$) from the cosine, and the cotangent ($$\cot$$) from the tangent.

## Unit circle

We can generalize these definitions at the unit circle for any angle. As the name implies, the radius of this circle has length one (unit length).

$$r^2 = x^2 + y^2 = \cos^2 \alpha + \sin^2 \alpha = 1$$

Thus, for any point on the circle, the $$x$$ component is the cosine (green), the $$y$$ component the sine (red). The tangent (dashed) is the ratio of sine to cosine.