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Trigonometric functions 2

By : wasin singhanan

Unit circle angles

The six trigonometric functions can also be defined in terms of the unit circle, the circle of radius one centered at the origin. The unit circle definition provides little in the way of practical calculation; indeed it relies on right triangles for most angles.

It also provides a single visual picture that encapsulates at once all the important triangles. From the Pythagorean theorem the equation for the unit circle is:

$x^2 + y^2 = 1. \,$

Inverse functions

Function	Definition	Value Field
$\arcsin x = y \,$	$\sin y = x \,$	$-\frac{\pi}{2} \le y \le \frac{\pi}{2} \,$
$\arccos x = y \,$	$\cos y = x \,$	$0 \le y \le \pi \,$
$\arctan x = y \,$	$\tan y = x \,$	$-\frac{\pi}{2} < y < \frac{\pi}{2} \,$
$\arccsc x = y \,$	$\csc y = x \,$	$-\frac{\pi}{2} \le y \le \frac{\pi}{2}, y \ne 0 \,$
$\arcsec x = y \,$	$\sec y = x \,$	$0 \le y \le \pi, y \ne \frac{\pi}{2} \,$
$\arccot x = y \,$	$\cot y = x \,$	$0 < y < \pi \,$

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Tag : Mathematics,

Trigonometric functions

By : wasin singhanan

Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications.

Right-angled triangle definitions

Function	Abbreviation	Description	Identities (using radians)
Sine	sin	opposite / hypotenuse	$\sin \theta = \cos \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\csc \theta}$
Cosine	cos	adjacent / hypotenuse	$\cos \theta = \sin \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\sec \theta}\,$
Tangent	tan (or tg)	opposite / adjacent	$\tan \theta = \frac{\sin \theta}{\cos \theta} = \cot \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\cot \theta}$
Cotangent	cot (or cotan or cotg or ctg or ctn)	adjacent / opposite	$\cot \theta = \frac{\cos \theta}{\sin \theta} = \tan \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\tan \theta}$
Secant	sec	hypotenuse / adjacent	$\sec \theta = \csc \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\cos \theta}$
Cosecant	csc (or cosec)	hypotenuse / opposite	$\csc \theta = \sec \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\sin \theta}$