Trigonometric functions 2

Posted by : wasin singhanan วันอังคารที่ 18 กันยายน พ.ศ. 2555

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