The trigonometric functions sine and cosine are defined in terms of the coordinates of points lying on the unit circle $x^{2} + y^{2} = 1$ .

Let $θ$ be an angle measured counter-clockwise from the positive x-axis. Let $P = (x, y)$ be the point of intersection of the terminal side of and the unit circle.

Then, the cosine of the angle $θ$ , $cos θ$ , is defined to be the x-coordinate of P.

$cos θ = x$

The sine of the angle $θ$ , $sin θ$ , is defined to be the y-coordinate of P.

$sin θ = y$

For example, if $θ = \frac{5 π}{6}$ , then the reference angle for $θ$ is

π - θ = π - \frac{5 π}{6} = \frac{π}{6} = 30 º

Hence, the reference triangle is a 30-60-90 right triangle (see figure below).

In a 30-60-90 right triangle with hypotenuse of length 1, the side opposite the 60º angle has length $\frac{\sqrt{3}}{2}$ , and the side adjacent the 30º angle has length $\frac{1}{2}$ .

Since the reference triangle is in the second quadrant, the x-coordinate is negative, and the y-coordinate is positive.

Therefore, the point P of intersection of the terminal side of $θ$ and the unit circle is $(- \frac{\sqrt{3}}{2}, \frac{1}{2})$ , so that $cos θ = - \frac{\sqrt{3}}{2} \approx - 0.8660$ and $sin θ = \frac{1}{2} = 0.5$ .

Now as this angle $θ$ changes so do the coordinates of the point P and thus the values of $cos θ$ and $sin θ$ also change. In this way the functions $f (x) = cos x$ and $g (x) = sin x$ are defined.

Here are their graphs.

The graph of

y = cos x

The graph of

y = sin x

They are periodic functions with period equal to $2 π$ , which means that every $2 π$ units their graphs repeat.
They have maximum values of 1 (called the amplitude), and minimum values of -1. Their domains are the set of all real numbers and their ranges are [-1,1].