Trigonometric graphs

For $y=\sin kx$ and $y=\cos kx$, the period is $\frac{360°}{|k|}$ ($k\ne 0$). For $y=\tan kx$, it is $\frac{180°}{|k|}$.

• $y=\sin 3x$: period 120°
• $y=\cos \frac{1}{2}x$: period 720°
• $y=\tan 2x$: period 90°, with asymptotes every 90° starting at $x=45°$

A larger $k$ squeezes the graph horizontally.

For $y=\sin (x+p)$, $y=\cos (x+p)$ and $y=\tan (x+p)$, the graph moves $p$ units left if $p>0$ and right if $p<0$.

Example. $y=\sin (x+45°)$ is $y=\sin x$ moved 45° left. Its maximum moves from $x=90°$ to $x=45°$, and its $x$-intercepts from $0°$ and $180°$ to $-45°$ and $135°$.

Shift the key points (intercepts, maximum and minimum) first, then draw the curve through them.

For $y=a\sin x+q$ (and similarly for cosine):

• Amplitude $=|a|$
• Vertical shift $q$ moves the graph up
• Range: $q-|a|\le y\le q+|a|$

Example. $y=2\cos x+1$ has amplitude 2 and range $-1\le y\le 3$. Its $x$-intercepts are where $\cos x=-\frac{1}{2}$: $x=120°$ and $x=240°$ in $[0°;360°]$.

Tangent graphs have no amplitude, because they have no maximum or minimum.

With two graphs on the same axes:

• Solutions of $f(x)=g(x)$ are the $x$-values of the intersection points.
• $f(x)>g(x)$ where $f$ is above $g$.
• $f(x)=k$ has as many solutions as the times the line $y=k$ cuts the graph.

Example. $f(x)=2\sin 2x$ and $g(x)=2\cos 2x$ meet where $\tan 2x=1$: $2x=45°+k\cdot 180°$, so $x=22,5°+k\cdot 90°$. In $[0°;180°]$ they meet at $22,5°$ and $112,5°$, and $f(x)>g(x)$ for $22,5°<x<112,5°$.