### Sine, Cosine and Tangent

#### Sine

##### Definition

Sine is an odd function and is periodic with period $2\pi$. The sine function has a domain of all real numbers, and its range is $-1\leq \sin x \leq 1$.

#### Cosine

##### Definition

Cosine is an even function function and is periodic with period $2\pi$. The cosine function has a domain of all real numbers, and its range is $-1\leq \cos x \leq 1$.

#### Tangent

##### Definition

The tangent function repeats in intervals of $\pi$ and has asymptotes every multiple of $\dfrac{n\pi}{2}$ for odd $n$. The tangent function has all real numbers except odd multiples of $90^{\circ}$ or $\pi/2$ in its domain, and its range contains all real numbers.

Tangent is defined to be $\tan x = \dfrac{\sin x}{\cos x}.$

#### Common Trig Ratios

Here is a table of common angles and the values of their corresponding trigonometric ratios.

Common Trigonometric Ratios

$x$

$0$

$\dfrac{\pi}{6}$

$\dfrac{\pi}{4}$

$\dfrac{\pi}{3}$

$\dfrac{\pi}{2}$

$\sin x$

$\;\;0\;\;$

$\;\,\dfrac{1}{2}$

$\dfrac{1}{\sqrt{2}}$

$\dfrac{\sqrt{3}}{2}$

$\;\;1\;\;$

$\cos x$

$\;\;1\;\;$

$\dfrac{\sqrt{3}}{2}$

$\dfrac{1}{\sqrt{2}}$

$\;\,\dfrac{1}{2}$

$\;\;0\;\;$

$\tan x$

$\;\;0\;\;$

$\dfrac{1}{\sqrt{3}}$

$\;\;1\;\;$

$\sqrt{3}$

$\;\infty\;$

#### Derivatives

Trig functions can be differentiated and integrated.

$f(x)$

$f'(x)$

$\sin x$

$\cos x$

$\cos x$

$-\sin x$

$\tan x$

$\sec^2 x$

#### Inverses

The inverses of these trigonometric functions are $\arcsin, \; \arccos, \; \arctan$, although they are also sometimes written as $\sin^{-1}, \; \cos^{-1}, \; \tan^{-1}$.

#### Video Examples

###### Example 1 - Sine

Prof. Robin Johnson sketches the graph of $\sin x$.

###### Example 2 - Tangent

Prof. Robin Johnson sketches the graph of $\tan x$.

#### Workbook

This workbook produced by HELM is a good revision aid, containing key points for revision and many worked examples.