### 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
$\sin x$ |
$\;\;0\;\;$ |
$\;\,\dfrac{1}{2}$ |
$\dfrac{1}{\sqrt{2}}$ |
$\dfrac{\sqrt{3}}{2}$ |
$\;\;1\;\;$ |

$\cos x$ |
$\;\;1\;\;$ |
$\dfrac{\sqrt{3}}{2}$ |
$\dfrac{1}{\sqrt{2}}$ |
$\;\,\dfrac{1}{2}$ |
$\;\;0\;\;$ |

$\tan x$ |
$\;\;0\;\;$ |
$\dfrac{1}{\sqrt{3}}$ |
$\;\;1\;\;$ |
$\sqrt{3}$ |
$\;\infty\;$ |

#### Derivatives

Trig functions can be differentiated and integrated.

$\sin x$ |
$\cos x$ |

$\cos x$ |
$-\sin x$ |

$\tan x$ |
$\sec^2 x$ |

#### 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.

#### See Also

#### External Resources