The unit circle turns sine and cosine into functions of a real angle. For every real $\theta$,

$$\cos\theta$$

is the horizontal coordinate of the corresponding point on the unit circle, while

$$\sin\theta$$

is the vertical coordinate.

Sine and cosine

Both functions have domain $\mathbb R$ and range $[-1,1]$. They repeat every full revolution:

$$\sin(\theta+2\pi)=\sin\theta,$$

$$\cos(\theta+2\pi)=\cos\theta.$$

Their graphs therefore repeat with period $2\pi$.

Tangent

Where $\cos\theta\ne0$, tangent is defined by

$$\tan\theta=\frac{\sin\theta}{\cos\theta}.$$

Because cosine is zero at angles such as $\pi/2$, tangent is undefined there. Tangent repeats every $\pi$ radians.

Transforming sinusoidal functions

A function written as

$$f(x)=A\sin(B(x-h))+D$$

is built from the basic sine graph. The amplitude is $|A|$, the horizontal shift is $h$, and the vertical shift is $D$. When $B\ne0$, the period is

$$\frac{2\pi}{|B|}.$$

Cosine transforms in the same way. These transformations make sine and cosine useful for describing repeating quantities such as oscillations, waves and alternating signals.