Trigonometric Identities

Fundamental Trigonometric Properties

Theorem: Trigonometric Functions of Standard Angles

Here are some common values for sine, cosine, tangent and cotangent:

	$0$	$\frac{\pi }{6}$	$\frac{\pi }{4}$	$\frac{\pi }{3}$	$\frac{\pi }{2}$	$\frac{2\pi }{3}$	$\frac{5\pi }{6}$	$\pi$
$\sin \theta$	$0$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$0$
$\cos \theta$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$	$-\frac{1}{2}$	$-\frac{\sqrt{3}}{2}$	$-1$
$\tan \theta$	$0$	$\frac{\sqrt{3}}{3}$	$1$	$\sqrt{3}$		$-\sqrt{3}$	$-\frac{\sqrt{3}}{3}$	$0$
$\cot \theta$		$\sqrt{3}$	$1$	$\frac{\sqrt{3}}{3}$	$0$	$-\frac{\sqrt{3}}{3}$	$-\sqrt{3}$

PROOF

TODO

Theorem: Fundamental Trigonometric Identities

Sine, cosine, tangent and cotangent obey the following identities:

$${\sin }^2\theta +{\cos }^2\theta =1$$ $$\tan \theta \cot \theta =1$$

PROOF

TODO

Theorem: Universal Trigonometric Substitution

The sine, cosine, tangent and cotangent of $\theta$ can all be expressed in terms of $\tan \frac{\theta }{2}$:

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

PROOF

TODO

Angle Sums

Theorem: Trigonometric Identities for Angle Sums

Sine, cosine, tangent and cotangent have the following properties:

$$\begin{array}{rl}\sin (\theta \pm \phi )& =\sin \theta \cos \phi \pm \cos \theta \sin \phi \\ & \\ \cos (\theta \pm \phi )& =\cos \theta \cos \phi \mp \sin \theta \phi \\ & \\ \tan (\theta \pm \phi )& =\frac{\tan \theta \pm \tan \phi }{1\mp \tan \theta \tan \phi }\\ & \\ \cot (\theta \pm \phi )& =\frac{\cot \theta \cot \phi \mp 1}{\cot \phi \pm \cot \theta }\end{array}$$

PROOF

TODO

Angle Products

Theorem: Chebyshev's Formulas

The sine, cosine and tangent obey Chebyshev’s formulas for every $r\in \mathbb{R}$:

$$\begin{array}{rl}\sin (r\phi )& =2\cos \phi \sin ((r-1)\phi )-\sin ((r-2)\phi )\\ & \\ \cos (r\phi )& =2\cos \phi \cos ((r-1)\phi )-\cos ((r-2)\phi )\\ & \\ \tan (r\phi )& =\frac{\tan ((r-1)\phi )+\tan \phi }{1-\tan \phi \tan ((r-1)\phi )}\end{array}$$

PROOF

TODO

Theorem: Double-Angle Formulas

The sine, cosine, tangent and cotangent of $2\theta$ can be expressed as

\begin{align*} \sin(2\theta) &= 2\sin \theta \cos \theta \\ \\ \cos(2\theta) &= 2\cos^2\theta + 1 = \cos^2 \theta - \sin^2 \theta = 1-2\sin^2 \theta \\ \\ \tan (2\theta) &= \frac{2\tan \theta}{1-\tan^2 \theta} \end{align*}

PROOF

TODO

Theorem: Half-Angle Formulae

The sine, cosine, tangent and cotangent obey the following properties:

$$\begin{array}{rl}\sin \frac{\theta }{2}& =\mathrm{sgn}\left(\sin \frac{\theta }{2}\right)\sqrt{\frac{1-\cos \theta }{2}}\\ & \\ \cos \frac{\theta }{2}& =\mathrm{sgn}\left(\cos \frac{\theta }{2}\right)\sqrt{\frac{1+\cos \theta }{2}}\\ & \\ \tan \frac{\theta }{2}& =\frac{\sin \theta }{1+\cos \theta }=\frac{1-\cos \theta }{\sin \theta }=\mathrm{sgn}(\sin \theta )\sqrt{\frac{1-\cos \theta }{1+\cos \theta }}\\ & \\ \cot \frac{\theta }{2}& =\frac{1+\cos \theta }{\sin \theta }=\frac{\sin \theta }{1-\cos \theta }=\pm \sqrt{\frac{1+\cos \theta }{1-\cos \theta }}\end{array}$$

PROOF

TODO

Argument Offsets

Theorem: Trigonometric Identities for Argument Offsets

The sine, cosine, tangent and cotangent have the following properties:

	$\frac{\pi }{2}+\theta$	$\frac{\pi }{2}-\theta$	$\pi +\theta$	$\pi -\theta$	$\frac{3\pi }{2}+\theta$	$\frac{3\pi }{2}-\theta$
$\sin$	$\cos \theta$	$\cos \theta$	$-\sin \theta$	$\sin \theta$	$-\cos \theta$	$-\cos \theta$
$\cos$	$-\sin \theta$	$\sin \theta$	$-\cos \theta$	$-\cos \theta$	$\sin \theta$	$-\sin \theta$
$\tan$	$-\cot \theta$	$\cot \theta$	$\tan \theta$	$-\tan \theta$	$-\cot \theta$	$\cot \theta$
$\cot$	$-\tan \theta$	$\tan \theta$	$\cot \theta$	$-\cot \theta$	$-\tan \theta$	$\tan \theta$

PROOF

TODO

Function Sums

Theorem: Trigonometric Identities for Function Sums

Sine, cosine, tangent and cotangent have the following properties:

$$\begin{array}{rl}\sin \theta \pm \sin \phi & =2\sin \frac{\theta \pm \phi }{2}\cos \frac{\theta \mp \phi }{2}\\ & \\ \cos \theta +\cos \phi & =2\cos \frac{\theta +\phi }{2}\cos \frac{\theta -\phi }{2}\\ & \\ \cos \theta -\cos \phi & =-2\sin \frac{\theta +\phi }{2}\sin \frac{\theta -\phi }{2}\\ & \\ \tan \theta \pm \tan \phi & =\frac{\sin (\theta \pm \phi )}{\cos \theta \cos \phi }\\ & \\ \cot \theta \pm \cot \phi & =\frac{\sin (\phi \pm \theta )}{\sin \theta \sin \phi }\end{array}$$

PROOF

TODO

Function Products

Theorem: Trigonometric Identities for Function Products

Sine, cosine, tangent and cotangent have the following properties:

$$\begin{array}{rl}\cos \theta \cos \phi & =\frac{1}{2}(\cos (\theta -\phi )+\cos (\theta +\phi ))\\ & \\ \sin \theta \sin \phi & =\frac{1}{2}(\cos (\theta -\phi )-\cos (\theta +\phi ))\\ & \\ \sin \theta \cos \phi & =\frac{1}{2}(\sin (\theta +\phi )+\sin (\theta -\phi ))\\ & \\ \cos \theta \sin \phi & =\frac{1}{2}(\sin (\theta +\phi )-\sin (\theta -\phi ))\\ & \\ \tan \theta \tan \phi & =\frac{\cos (\theta -\phi )-\cos (\theta +\phi )}{\cos (\theta -\phi )+\cos (\theta +\phi )}\\ & \\ \tan \theta \cot \phi & =\frac{\sin (\theta +\phi )+\sin (\theta -\phi )}{\sin (\theta +\phi )-\sin (\theta -\phi )}\end{array}$$

PROOF

TODO

Compositions

Theorem: Functions of Arcfunctions

The compositions of real inverse trigonometric functions inside Real Trigonometric Functions have the following properties:

$$\begin{array}{rl}& \cos (\arctan (x))=\frac{1}{\sqrt{1+x^2}}\\ & \cos (\arcsin (x))=\frac{1}{\sqrt{1-x^2}}\\ & \cos \left(\arctan \left(\frac{1}{x}\right)\right)=\frac{|x|}{\sqrt{1+x^2}}\end{array}$$

PROOF

TODO