Trigonometry/Natural trigonometric functions

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Note: Some values in the table are given in forms that include a radical in the denominator — this is done both to simplify recognition of reciprocal pairs and because the form given in the table is simpler in some sense. Note also that all absolute values of trigonometric functions for remarquable points that are listed in this table are contained in the first quadrant (from 0 to 90° or $π / 2$ radians, inclusive); all others are deduced by simple symetries with the horizontal or vertical axis, or by swapping axis (on the trigonometric circle) so that one trigonometric function is also swapped with its co-function.

$\theta\,$ (positive)		$\sin\theta\,$	$\cos\theta\,$	$\tan\theta\,$	$\cot\theta\,$	$\sec\theta\,$	$\csc\theta\,$	$\theta\,$ (negative)
(degrees)	(radians)	$\sin\theta\,$	$\cos\theta\,$	$\tan\theta\,$	$\cot\theta\,$	$\sec\theta\,$	$\csc\theta\,$	(degrees)	(radians)
0°	$0\,$	$0\,$	$1\,$	$0\,$	not defined	$1\,$	not defined	−360°	$-2\pi\,$
15°	$\frac{\pi}{12}\,$	$\frac{\sqrt{6}-\sqrt{2}}{4}\,$	$\frac{\sqrt{6}+\sqrt{2}}{4}\,$	$2-\sqrt{3}\,$	$2+\sqrt{3}\,$	$\frac{4}{\sqrt{6}+\sqrt{2}}\,$	$\frac{4}{\sqrt{6}-\sqrt{2}}\,$	−345°	$-\frac{13\pi}{12}\,$
22.5°	$\frac{\pi}{8}\,$	$\frac{\sqrt{2-\sqrt{2}}}{2}\,$	$\frac{\sqrt{2+\sqrt{2}}}{2}\,$	$\sqrt{2}-1\,$	$\sqrt{2}+1\,$	$\frac{2}{\sqrt{2+\sqrt{2}}}\,$	$\frac{2}{\sqrt{2-\sqrt{2}}}\,$	−337.5°	$-\frac{15\pi}{8}\,$
30°	$\frac{\pi}{6}\,$	$\frac{1}{2}\,$	$\frac{\sqrt{3}}{2}\,$	$\frac{1}{\sqrt{3}}\,$	$\sqrt{3}\,$	$\frac{2}{\sqrt{3}}\,$	$2\,$	−330°	$-\frac{11\pi}{6}\,$
45°	$\frac{\pi}{4}\,$	$\frac{1}{\sqrt{2}}\,$	$\frac{1}{\sqrt{2}}\,$	$1\,$	$1\,$	$\sqrt{2}\,$	$\sqrt{2}\,$	−315°	$-\frac{7\pi}{4}\,$
60°	$\frac{\pi}{3}\,$	$\frac{\sqrt{3}}{2}\,$	$\frac{1}{2}\,$	$\sqrt{3}\,$	$\frac{1}{\sqrt{3}}\,$	$2\,$	$\frac{2}{\sqrt{3}}\,$	−300°	$-\frac{5\pi}{3}\,$
67.5°	$\frac{3\pi}{8}\,$	$\frac{\sqrt{2+\sqrt{2}}}{2}\,$	$\frac{\sqrt{2-\sqrt{2}}}{2}\,$	$\sqrt{2}+1\,$	$\sqrt{2}-1\,$	$\frac{2}{\sqrt{2-\sqrt{2}}}\,$	$\frac{2}{\sqrt{2+\sqrt{2}}}\,$	−292.5°	$-\frac{11\pi}{8}\,$
75°	$\frac{5\pi}{12}\,$	$\frac{\sqrt{6}+\sqrt{2}}{4}\,$	$\frac{\sqrt{6}-\sqrt{2}}{4}\,$	$2+\sqrt{3}\,$	$2-\sqrt{3}\,$	$\frac{4}{\sqrt{6}-\sqrt{2}}\,$	$\frac{4}{\sqrt{6}+\sqrt{2}}\,$	−285°	$-\frac{19\pi}{12}\,$
90°	$\frac{\pi}{2}\,$	$1\,$	$0\,$	not defined	$0\,$	not defined	$1\,$	−270°	$-\frac{3\pi}{2}\,$
105°	$\frac{7\pi}{12}\,$	$\frac{\sqrt{6}+\sqrt{2}}{4}\,$	$-\frac{\sqrt{6}-\sqrt{2}}{4}\,$	$-2-\sqrt{3}\,$	$-2+\sqrt{3}\,$	$-\frac{4}{\sqrt{6}-\sqrt{2}}\,$	$\frac{4}{\sqrt{6}+\sqrt{2}}\,$	−255°	$-\frac{17\pi}{12}\,$
112.5°	$\frac{5\pi}{8}\,$	$\frac{\sqrt{2+\sqrt{2}}}{2}\,$	$-\frac{\sqrt{2-\sqrt{2}}}{2}\,$	$-\sqrt{2}-1\,$	$-\sqrt{2}+1\,$	$-\frac{2}{\sqrt{2-\sqrt{2}}}\,$	$\frac{2}{\sqrt{2+\sqrt{2}}}\,$	−247.5°	$-\frac{11\pi}{8}\,$
120°	$\frac{2\pi}{3}\,$	$\frac{\sqrt{3}}{2}\,$	$-\frac{1}{2}\,$	$-\sqrt{3}\,$	$-\frac{1}{\sqrt{3}}\,$	$-2\,$	$\frac{2}{\sqrt{3}}\,$	−240°	$-\frac{4\pi}{3}\,$
135°	$\frac{3\pi}{4}\,$	$\frac{1}{\sqrt{2}}\,$	$-\frac{1}{\sqrt{2}}\,$	$-1\,$	$-1\,$	$-\sqrt{2}\,$	$\sqrt{2}\,$	−225°	$-\frac{5\pi}{4}\,$
150°	$\frac{5\pi}{6}\,$	$\frac{1}{2}\,$	$-\frac{\sqrt{3}}{2}\,$	$-\frac{1}{\sqrt{3}}\,$	$-\sqrt{3}\,$	$-\frac{2}{\sqrt{3}}\,$	$2\,$	−210°	$-\frac{7\pi}{6}\,$
157.5°	$\frac{7\pi}{8}\,$	$\frac{\sqrt{2-\sqrt{2}}}{2}\,$	$-\frac{\sqrt{2+\sqrt{2}}}{2}\,$	$-\sqrt{2}+1\,$	$-\sqrt{2}-1\,$	$-\frac{2}{\sqrt{2+\sqrt{2}}}\,$	$\frac{2}{\sqrt{2-\sqrt{2}}}\,$	−202.5°	$-\frac{9\pi}{8}\,$
165°	$\frac{11\pi}{12}\,$	$\frac{\sqrt{6}-\sqrt{2}}{4}\,$	$-\frac{\sqrt{6}+\sqrt{2}}{4}\,$	$-2+\sqrt{3}\,$	$-2-\sqrt{3}\,$	$-\frac{4}{\sqrt{6}+\sqrt{2}}\,$	$\frac{4}{\sqrt{6}-\sqrt{2}}\,$	−195°	$-\frac{13\pi}{12}\,$
180°	$\pi\,$	$0\,$	$-1\,$	$0\,$	not defined	$-1\,$	not defined	−180°	$-\pi\,$
195°	$\frac{13\pi}{12}\,$	$-\frac{\sqrt{6}-\sqrt{2}}{4}\,$	$-\frac{\sqrt{6}+\sqrt{2}}{4}\,$	$2-\sqrt{3}\,$	$2+\sqrt{3}\,$	$-\frac{4}{\sqrt{6}+\sqrt{2}}\,$	$-\frac{4}{\sqrt{6}-\sqrt{2}}\,$	−165°	$-\frac{11\pi}{12}\,$
202.5°	$\frac{9\pi}{8}\,$	$-\frac{\sqrt{2-\sqrt{2}}}{2}\,$	$-\frac{\sqrt{2+\sqrt{2}}}{2}\,$	$\sqrt{2}-1\,$	$\sqrt{2}+1\,$	$-\frac{2}{\sqrt{2+\sqrt{2}}}\,$	$-\frac{2}{\sqrt{2-\sqrt{2}}}\,$	−157.5°	$-\frac{7\pi}{8}\,$
210°	$\frac{7\pi}{6}\,$	$-\frac{1}{2}\,$	$-\frac{\sqrt{3}}{2}\,$	$\frac{1}{\sqrt{3}}\,$	$\sqrt{3}\,$	$-\frac{2}{\sqrt{3}}\,$	$-2\,$	−150°	$-\frac{5\pi}{6}\,$
225°	$\frac{5\pi}{4}\,$	$-\frac{1}{\sqrt{2}}\,$	$-\frac{1}{\sqrt{2}}\,$	$1\,$	$1\,$	$-\sqrt{2}\,$	$-\sqrt{2}\,$	−135°	$-\frac{3\pi}{4}\,$
240°	$\frac{4\pi}{3}\,$	$-\frac{\sqrt{3}}{2}\,$	$-\frac{1}{2}\,$	$\sqrt{3}\,$	$\frac{1}{\sqrt{3}}\,$	$-2\,$	$-\frac{2}{\sqrt{3}}\,$	−120°	$-\frac{2\pi}{3}\,$
247.5°	$\frac{11\pi}{8}\,$	$-\frac{\sqrt{2+\sqrt{2}}}{2}\,$	$-\frac{\sqrt{2-\sqrt{2}}}{2}\,$	$\sqrt{2}+1\,$	$\sqrt{2}-1\,$	$-\frac{2}{\sqrt{2-\sqrt{2}}}\,$	$-\frac{2}{\sqrt{2+\sqrt{2}}}\,$	−112.5°	$-\frac{5\pi}{8}\,$
255°	$\frac{17\pi}{12}\,$	$-\frac{\sqrt{6}+\sqrt{2}}{4}\,$	$-\frac{\sqrt{6}-\sqrt{2}}{4}\,$	$2+\sqrt{3}\,$	$2-\sqrt{3}\,$	$-\frac{4}{\sqrt{6}-\sqrt{2}}\,$	$-\frac{4}{\sqrt{6}+\sqrt{2}}\,$	−105°	$-\frac{7\pi}{12}\,$
270°	$\frac{3\pi}{2}\,$	$-1\,$	$0\,$	not defined	$0\,$	not defined	$-1\,$	−90°	$-\frac{\pi}{2}\,$
285°	$\frac{19\pi}{12}\,$	$-\frac{\sqrt{6}+\sqrt{2}}{4}\,$	$\frac{\sqrt{6}-\sqrt{2}}{4}\,$	$-2-\sqrt{3}\,$	$-2+\sqrt{3}\,$	$\frac{4}{\sqrt{6}-\sqrt{2}}\,$	$-\frac{4}{\sqrt{6}+\sqrt{2}}\,$	−75°	$-\frac{5\pi}{12}\,$
292.5°	$\frac{11\pi}{8}\,$	$-\frac{\sqrt{2+\sqrt{2}}}{2}\,$	$\frac{\sqrt{2-\sqrt{2}}}{2}\,$	$-\sqrt{2}-1\,$	$-\sqrt{2}+1\,$	$\frac{2}{\sqrt{2-\sqrt{2}}}\,$	$-\frac{2}{\sqrt{2+\sqrt{2}}}\,$	−67.5°	$-\frac{3\pi}{8}\,$
300°	$\frac{5\pi}{3}\,$	$-\frac{\sqrt{3}}{2}\,$	$\frac{1}{2}\,$	$-\sqrt{3}\,$	$-\frac{1}{\sqrt{3}}\,$	$2\,$	$-\frac{2}{\sqrt{3}}\,$	−60°	$-\frac{\pi}{3}\,$
315°	$\frac{7\pi}{4}\,$	$-\frac{1}{\sqrt{2}}\,$	$\frac{1}{\sqrt{2}}\,$	$-1\,$	$-1\,$	$\sqrt{2}\,$	$-\sqrt{2}\,$	−45°	$-\frac{\pi}{4}\,$
330°	$\frac{11\pi}{6}\,$	$-\frac{1}{2}\,$	$\frac{\sqrt{3}}{2}\,$	$-\frac{1}{\sqrt{3}}\,$	$-\sqrt{3}\,$	$\frac{2}{\sqrt{3}}\,$	$-2\,$	−30°	$-\frac{\pi}{6}\,$
337.5°	$\frac{15\pi}{8}\,$	$-\frac{\sqrt{2-\sqrt{2}}}{2}\,$	$\frac{\sqrt{2+\sqrt{2}}}{2}\,$	$-\sqrt{2}+1\,$	$-\sqrt{2}-1\,$	$\frac{2}{\sqrt{2+\sqrt{2}}}\,$	$-\frac{2}{\sqrt{2-\sqrt{2}}}\,$	−22.5°	$-\frac{\pi}{8}\,$
345°	$\frac{13\pi}{12}\,$	$-\frac{\sqrt{6}-\sqrt{2}}{4}\,$	$\frac{\sqrt{6}+\sqrt{2}}{4}\,$	$-2+\sqrt{3}\,$	$-2-\sqrt{3}\,$	$\frac{4}{\sqrt{6}+\sqrt{2}}\,$	$-\frac{4}{\sqrt{6}-\sqrt{2}}\,$	−15°	$-\frac{\pi}{12}\,$
360°	$2\pi\,$	$0\,$	$1\,$	$0\,$	not defined	$1\,$	not defined	0°	$0\,$

Notice that for certain values of $x\,$ , the tangent, cotangent, secant, and cosecant functions are undefined. This is because these functions are defined as $\frac{\sin x}{\cos x}\,$ , $\frac{\cos x}{\sin x}\,$ , $\frac1{\cos x}\,$ , and $\frac1{\sin x}\,$ , respectively. Since an expression is undefined if it is divided by zero, the functions are therefore undefined at angle measures where the denominator (the sine or cosine of $x\,$ , depending on the trigonometric function) is equal to zero. Take, for example, the tangent function. If the tangent function is analyzed at 90 degrees ( $\pi/2\,$ radians), the function is then equivalent to $\frac{\sin(\pi/2)}{\cos(\pi/2)}\,$ , or $\frac10\,$ , which is an undefined value.