Connections of trigonometric functions in tables: reduction formulas, addition theorems, ...

Contexts of trigonometric functions

Trigonometry

Properties of trigonometric functions

Representation by other trigonometric functions

	$\sin^{2} x$	$\cos^{2} x$	$\tan^{2} x$	$\cot^{2} x$
$\sin^{2} x$	-	$1 - \cos^{2} x$	$\frac{\tan^{2} x}{1 + \tan^{2} x}$	$\frac{1}{1 + \cot^{2} x}$
$\cos^{2} x$	$1 - \sin^{2} x$	-	$\frac{1}{1 + \tan^{2} x}$	$\frac{\cot^{2} x}{1 + \cot^{2} x}$
$\tan^{2} x$	$\frac{\sin^{2} x}{1 - \sin^{2} x}$	$\frac{1 - \cos^{2} x}{\cos^{2} x}$	-	$\frac{1}{\cot^{2} x}$
$\cot^{2} x$	$\frac{1 - \sin^{2} x}{\sin^{2} x}$	$\frac{\cos^{2} x}{1 - \cos^{2} x}$	$\frac{1}{\tan^{2} x}$	-

Functional values of trigonometric functions

Radian	0	$\frac{π}{6}$	$\frac{π}{4}$	$\frac{π}{3}$	$\frac{π}{2}$
Degree	0°	30°	45°	60°	90°
$\sin^{2} x$	0	$\frac{1}{2}$	$\frac{1}{2} \sqrt{2}$	$\frac{1}{2} \sqrt{3}$	1
$\cos^{2} x$	1	$\frac{1}{2} \sqrt{3}$	$\frac{1}{2} \sqrt{2}$	$\frac{1}{2}$	0
$\tan^{2} x$	0	$\frac{1}{3} \sqrt{3}$	1	$\sqrt{3}$	-
$\cot^{2} x$	-	$\sqrt{3}$	1	$\frac{1}{3} \sqrt{3}$	0

Reduction formulas (in degrees)

$\sin (90° + x)$ $= \cos (x)$

$\cos (90° + x)$ $= - \sin (x)$

$\tan (90° + x)$ $= - \cot (x)$

$\cot (90° + x)$ $= - \tan (x)$

$\sin (180° + x)$ $= - \sin (x)$

$\cos (180° + x)$ $= - \cos (x)$

$\tan (180° + x)$ $= \tan (x)$

$\cot (180° + x)$ $= \cot (x)$

Reduction formulas (in radians)

$\sin (\frac{π}{2} + x)$ $= \cos (x)$

$\cos (\frac{π}{2} + x)$ $= - \sin (x)$

$\tan (\frac{π}{2} + x)$ $= - \cot (x)$

$\cot (\frac{π}{2} + x)$ $= - \tan (x)$

$\sin (π + x)$ $= - \sin (x)$

$\cos (π + x)$ $= - \cos (x)$

$\tan (π + x)$ $= \tan (x)$

$\cot (π + x)$ $= \cot (x)$

Context of the trigonometric functions for the same argument

${\sin (x)}^{2} + {\cos (x)}^{2}$ $= 1$

$\tan (x)$ $= \frac{\sin (x)}{\cos (x)}$

$\cot (x)$ $= \frac{\cos (x)}{\sin (x)}$ $= \frac{1}{\tan (x)}$

$\cot (x) \cdot \tan (x)$ $= 1$

Addition theorems of trigonometric functions

$\sin (x \pm y)$ $= \sin (x) \cos (y) \pm \cos (x) \sin (y)$

$\cos (x \pm y)$ $= \cos (x) \cos (y) \mp \sin (x) \sin (y)$

$\tan (x \pm y)$ $= \frac{\tan (x) \pm \tan (y)}{1 \mp \tan (x) \tan (y)}$

Addition theorems for multiples of the argument value

$\sin (2 x)$ $= 2 \sin (x) \cos (x)$

$\sin (3 x)$ $= 3 \sin (x) - 4 \sin^{3} (x)$

$\cos (2 x)$ $= \cos^{2} (x) - \sin^{2} (x)$

$\cos (3 x)$ $= 4 \cos^{3} (x) - 3 \cos (x)$

Half of the argument value

$\sin (\frac{x}{2})$ $= \pm \sqrt{\frac{1 - \cos (x)}{2}}$

$\cos (\frac{x}{2})$ $= \pm \sqrt{\frac{1 + \cos (x)}{2}}$

$\tan (\frac{x}{2})$ $= \pm \sqrt{\frac{1 - \cos (x)}{1 + \cos (x)}}$

Sum and difference of trigonometric functions

$\sin (x) + \sin (y)$ $= 2 \sin (\frac{x + y}{2}) \cos (\frac{x - y}{2})$

$\cos (x) + \cos (y)$ $= 2 \cos (\frac{x + y}{2}) \cos (\frac{x - y}{2})$

$\sin (x) - \sin (y)$ $= 2 \cos (\frac{x + y}{2}) \sin (\frac{x - y}{2})$

$\cos (x) - \cos (y)$ $= - 2 \sin (\frac{x + y}{2}) \sin (\frac{x - y}{2})$

$\cos (x) \pm \sin (x)$ $= \sqrt{2} \sin (\frac{π}{4} \pm x)$ $= \sqrt{2} \cos (\frac{π}{4} \mp x)$

$\tan (x) \pm \tan (y)$ $= \frac{\sin (x \pm y)}{\cos (x) \cos (y)}$

$\cot (x) \pm \cot (y)$ $= \pm \frac{\sin (x \pm y)}{\sin (x) \sin (y)}$

$\tan (x) + \cot (y)$ $= \frac{\cos (x - y)}{\cos (x) \sin (y)}$

$\cot (x) - \tan (y)$ $= \frac{\cos (x + y)}{\sin (x) \cos (y)}$

Products of trigonometric functions

$\sin (x) \sin (y)$ $= \frac{1}{2} (\cos (x - y) - \cos (x + y))$

$\cos (x) \cos (y)$ $= \frac{1}{2} (\cos (x - y) + \cos (x + y))$

$\sin (x) \cos (y)$ $= \frac{1}{2} (\sin (x - y) + \sin (x + y))$

$\cos (x) \sin (y)$ $= \frac{1}{2} (\sin (x + y) - \sin (x - y))$

Potencies of trigonometric functions

$\sin^{2} (x)$ $= \frac{1}{2} (1 - \cos (2 x))$

$\sin^{3} (x)$ $= \frac{1}{4} (3 \sin (x) - \sin (3 x))$

$\sin^{n} (x)$ $= \frac{1}{2^{n}} \sum_{k = 0}^{n} (\begin{matrix} n \\ k \end{matrix}) \cos ((n - 2 k) (x - \frac{π}{2}))$

$\cos^{2} (x)$ $= \frac{1}{2} (1 + \cos (2 x))$

$\cos^{3} (x)$ $= \frac{1}{4} (3 \cos (x) + \cos (3 x))$

$\cos^{n} (x)$ $= \frac{1}{2^{n}} \sum_{k = 0}^{n} (\begin{matrix} n \\ k \end{matrix}) \cos ((n - 2 k) x)$