The graphs of sin, cos and tan

Unroll the circle

The unit circle is the circle with centre $(0, 0)$ and radius 1.

Put a point $P$ on it. Measure the angle $x$ anticlockwise from the positive $x$ -axis to the radius $OP$ . (Here $x$ is the angle, so we call the $x$ -coordinate the "sideways position" to avoid a clash.) For that point:

its height ( $y$ -coordinate) is $\sin x$
its sideways position ( $x$ -coordinate) is $\cos x$
height $\div$ sideways position is $\tan x$

Now walk $P$ round the circle and plot each of these as the angle $x$ grows:

the height traces out the wave $y = \sin x$
the sideways position traces out the wave $y = \cos x$
the ratio traces $y = \tan x$ , which blows up whenever the sideways position is zero (you can't divide by 0)

Every lap of the circle repeats the pattern. That repeat length is called the period.

Key features

Words you need for the table:

Range: all the $y$ -values the graph reaches.
Zeros: where the graph crosses the $x$ -axis ( $y = 0$ ).
Asymptote: a line the graph gets closer and closer to but never touches. For tan, these are vertical lines where $\cos x = 0$ .

Read the table one column at a time:

	$y = \sin x$	$y = \cos x$	$y = \tan x$
Period	$360^\circ$ ( $2\pi$ )	$360^\circ$ ( $2\pi$ )	$180^\circ$ ( $\pi$ )
Range	$-1 \leqslant y \leqslant 1$	$-1 \leqslant y \leqslant 1$	all real $y$
Zeros ( $0^\circ$ to $360^\circ$ )	$0^\circ, 180^\circ, 360^\circ$	$90^\circ, 270^\circ$	$0^\circ, 180^\circ, 360^\circ$
Maximum $1$ at	$90^\circ$	$0^\circ, 360^\circ$	none
Minimum $-1$ at	$270^\circ$	$180^\circ$	none
Asymptotes	none	none	$x = 90^\circ, 270^\circ$

Beyond $0^\circ$ to $360^\circ$ , keep repeating: the graphs go on forever in both directions.