Graphs of sin, cos and tan

In $y=\sin x$, the horizontal graph coordinate $x$ is now the angle, not the horizontal coordinate of the point on the circle. The output is the point's vertical coordinate.

At quarter turns, sine follows 0, 1, 0, −1, 0. Cosine follows the circle's horizontal coordinate: 1, 0, −1, 0, 1. Plot these points and join them with a smooth curve.

A period is the smallest positive angle shift that repeats the whole graph. Sine and cosine have period $360^\circ$ or $2\pi$. Both have domain all real angles and range $[-1,1]$, meaning every output from −1 to 1.

The midline lies halfway between the maximum and minimum; here it is $y=0$. The amplitude is the distance from the midline to a maximum; here it is 1, not the full height 2.

Tangent is sine divided by cosine. Near 90°, cosine approaches zero while sine approaches 1, so the quotient grows without bound. The vertical line $x=90^\circ$ is an asymptote: the curve approaches it, but no tangent value exists there.

The same happens at 270° and after each extra 180°. Draw dashed asymptotes first; never join a tangent curve across them. Between asymptotes each branch rises. Tangent can output any real number, so it has no maximum, minimum or amplitude.

A half turn changes both sine and cosine signs, leaving their quotient unchanged. That is why tangent's period is $180^\circ$ or $\pi$, not $2\pi$.

For a longer or negative interval, continue the same repeating pattern. Useful symmetry checks are $\sin(-x)=-\sin x$, $\cos(-x)=\cos x$ and $\tan(-x)=-\tan x$, where tangent is defined.