Graphs of Trigonometric functions

Graphs of Trigonometric functions

All trigonometric functions are periodic, with sine and cosine repeating every $2\pi$ radians and tangent repeating every $\pi$ radians. Plotting them over one period turns these algebraic facts into a picture.

i) Graph of $y=\sin x$, for $-\pi<x<\pi$. Sample values (with the negative-x values obtained from $\sin(-\theta)=-\sin\theta$):

Plotting the x-axis horizontally and y-axis vertically, the curve rises smoothly from 0 at $x=0$ to a peak of 1 at $x=\pi/2$, then falls back through 0 at $x=\pi$; by symmetry it continues to a trough of $-1$ near $x=-\pi/2$. The whole curve stays within one unit of the x-axis (confirming range $[-1,1]$) and, since the period is $2\pi$, shifting this curve left or right by $2\pi$ makes it fall exactly back onto itself.

ii) Graph of $y=\cos x$, for $-\pi<x<\pi$. Sample values (using $\cos(-\theta)=\cos\theta$):

The cosine curve has the identical wave shape to the sine curve, but starts at its peak value 1 at $x=0$ rather than at 0 — it is the sine wave shifted left by $\pi/2$. It too stays within one unit of the x-axis and repeats every $2\pi$.

iii) Graph of $y=\tan x$, for $-\pi/2<x<\pi/2$. tanx does not exist at $x=\pm\pi/2$. As x increases from 0 towards $\pi/2$, $\sin x$ increases from 0 to 1 while $\cos x$ decreases from 1 to 0, so $\tan x=\sin x/\cos x$ increases without bound; by symmetry it decreases without bound as x approaches $-\pi/2$.