Domain and Range of Trigonometric Functions

For a periodic real-valued function $f$, a number $p>0$ is a period if $x+p$ lies in the domain of $f$ whenever $x$ does, and $f(x+p)=f(x)$ for every such $x$; the smallest such $p$ is called the period of $f$.

• $\sin x,\cos x,\sec x,\text{cosec}\,x$ (and, more generally, $e^{ix}$) are periodic with period $2\pi$.
• $\tan x,\cot x$ are periodic with the shorter period $\pi$ — halving the period is possible precisely because $\tan(x+\pi)=\tan x$ even though $\tan(x+2\pi)$ is also true; $\pi$ is the smallest number for which the identity holds.

Odd and even functions. $f$ is even if $-x$ is in the domain whenever $x$ is, and $f(-x)=f(x)$ for all such $x$; $f$ is odd if instead $f(-x)=-f(x)$. Among the six trig functions: $\sin x,\tan x,\cot x,\text{cosec}\,x$ (and $x^3$) are odd; $\cos x,\sec x$ (and $x^2$) are even.

Combining periods. If $f=g\pm h$ where $g,h$ are periodic with periods $p_g,p_h$, then $f$'s period is $\text{lcm}(p_g,p_h)$ whenever that lcm exists. For instance, $y=\cos6x+\sin4x$ has period $\text{lcm}\left(\tfrac{2\pi}6,\tfrac{2\pi}4\right)=\text{lcm}\left(\tfrac{\pi}3,\tfrac{\pi}2\right)=\pi$, and $y=\cos x-\sin x$ has period $\text{lcm}(2\pi,2\pi)=2\pi$.

Table — domain and range of the six trigonometric functions: