# Trigonometry

## Trigonometric Graphs

We can draw the graphs of the trigonometric functions in the coordinate plane as follows. (See the Graphing section for graphs of other functions.)
Domain Range Period
$\sin$ $\mathbb{R}$ $[-1,1]$ $2\pi$
$\cos$ $\mathbb{R}$ $[-1,1]$ $2\pi$

The graph of the function $y=\sin(x)$

The graph of the function $y=\cos(x)$

Note: The graphs of $y=\sin\left(x\right)$ and $y=\cos\left(x\right)$ are the same except one is a shifted version of the other. For this reason, both their graphs and transformations are called sinusoidal functions: $$y=a \sin\left(cx-b\right) + d \hspace{30 mm} y=a \cos\left(cx-b\right) + d$$ where $a$ is the amplitude, $b/c$ is the phase shift, the domain is $\mathbb{R}$ and the range is $[-a+d,a+d]$.

Domain Range Period
$\tan$ $\mathbb{R}\setminus \{k\pi/2\}, k=\{\pm 1,\pm 3,\ldots\}$ $\mathbb{R}$ $\pi$
$\cot$ $\mathbb{R}\setminus \{k\pi\}, k \in \mathbb{Z}$ $\mathbb{R}$ $\pi$

The graph of the function $y=\tan(x)$

The graph of the function $y=\cot(x)$

Domain Range Period
$\sec$ $\mathbb{R}\setminus \{k\pi/2\}, k=\{\pm 1,\pm 3,\ldots\}$ $\left(-\infty,-1] \cup [1,\infty\right)$ $2\pi$
$\csc$ $\mathbb{R}\setminus \{k\pi\}, k \in \mathbb{Z}$ $\left(-\infty,-1] \cup [1,\infty\right)$ $2\pi$

The graph of the function $y=\sec(x)$

The graph of the function $y=\csc(x)$