Trigonometric Function Graphs

Lesson · Beginner

Trigonometry

Function	Domain	Range	Period	Zeros	Vertical Asymptotes	Symmetry
$\sin x$	$\mathbb{R}$	$[-1,1]$	$2\pi$	$k\pi$	None	Odd
$\cos x$	$\mathbb{R}$	$[-1,1]$	$2\pi$	$\frac{\pi}{2}+k\pi$	None	Even
$\tan x$	$x\ne\frac{\pi}{2}+k\pi$	$\mathbb{R}$	$\pi$	$k\pi$	$\frac{\pi}{2}+k\pi$	Odd
$\cot x$	$x\ne k\pi$	$\mathbb{R}$	$\pi$	$\frac{\pi}{2}+k\pi$	$k\pi$	Odd

where

k\in\mathbb Z

Assume

a\ne0,\qquad b\ne0

Function	Amplitude	Period	Phase Shift	Vertical Shift	Midline
$y=a\sin(bx+c)+d$	$\|a\|$	$\frac{2\pi}{\|b\|}$	$-\frac{c}{b}$	$d$	$y=d$
$y=a\cos(bx+c)+d$	$\|a\|$	$\frac{2\pi}{\|b\|}$	$-\frac{c}{b}$	$d$	$y=d$
$y=a\tan(bx+c)+d$	None	$\frac{\pi}{\|b\|}$	$-\frac{c}{b}$	$d$	$y=d$
$y=a\cot(bx+c)+d$	None	$\frac{\pi}{\|b\|}$	$-\frac{c}{b}$	$d$	$y=d$

Function	Domain	Range	Period	Zeros	Vertical Asymptotes	Symmetry
$a\sin(bx+c)+d$	$\mathbb{R}$	$[d-\|a\|,d+\|a\|]$	$\frac{2\pi}{\|b\|}$	$x=\frac{k\pi-c}{b}\text{ if }d=0$	None	Not necessarily even or odd
$a\cos(bx+c)+d$	$\mathbb{R}$	$[d-\|a\|,d+\|a\|]$	$\frac{2\pi}{\|b\|}$	$x=\frac{\frac{\pi}{2}+k\pi-c}{b}\text{ if }d=0$	None	Not necessarily even or odd
$a\tan(bx+c)+d$	$x\ne\frac{\frac{\pi}{2}+k\pi-c}{b}$	$\mathbb{R}$	$\frac{\pi}{\|b\|}$	$x=\frac{\arctan(-d/a)+k\pi-c}{b}$	$x=\frac{\frac{\pi}{2}+k\pi-c}{b}$	Point symmetry about its centers
$a\cot(bx+c)+d$	$x\ne\frac{k\pi-c}{b}$	$\mathbb{R}$	$\frac{\pi}{\|b\|}$	$x=\frac{\operatorname{arccot}(-d/a)+k\pi-c}{b}$	$x=\frac{k\pi-c}{b}$	Point symmetry about its centers