From 561bac75579391c14e47eaccfabdf9eda98855da Mon Sep 17 00:00:00 2001
From: Thomas Albers <thomas@thomaslabs.org>
Date: Wed, 27 Jul 2022 18:13:20 +0200
Subject: Initial commit

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 math/trigonometry.org | 51 +++++++++++++++++++++++++++++++++++++++++++++++++++
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+#+TITLE:Trigonometric identities
+#+SETUPFILE: ../math_options.org
+#+LATEX_HEADER: \usepackage{bm}
+#+LATEX_HEADER: \usepackage{mathtools}
+
+* Disclaimer
+This site as of now just a technology demonstration and its claims
+should not be taken as true (even though I myself am pretty confident
+they are)
+
+* Pythagorean identities
+\begin{align*}
+&\cos^2\left(x\right) + \sin^2\left(x\right) = 1\\
+&\tan^2\left(x\right) + 1                    = \sec^2\left(x\right)\\
+&1                    + \cot^2\left(x\right) = \csc^2\left(x\right)
+\end{align*}
+* Sum of angles
+\begin{align*}
+&\sin\left(a \pm b\right) = \sin\left(a\right)\cos\left(b\right) \pm \cos\left(a\right)\sin\left(b\right)\\
+&\cos\left(a \pm b\right) = \cos\left(a\right)\cos\left(b\right) \mp \sin\left(a\right)\sin\left(b\right)\\
+&\tan(a \pm b) = \frac{\tan(a) \pm \tan(b)}{1 \mp \tan(a) \tan(b)}
+\end{align*}
+* Multiple angles
+\begin{align*}
+&\sin(2\theta) = 2\sin(\theta)\cos(\theta)\\
+&\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta
+\end{align*}
+* Half-angle formulae
+\begin{align*}
+&\sin \frac{\theta}{2} = \pm \sqrt{\frac{1-\cos\theta}{2}}\\
+&\cos \frac{\theta}{2} = \pm \sqrt{\frac{1+\cos\theta}{2}}
+\end{align*}
+* Power-reduction formulae
+\begin{align*}
+&\sin^2\theta = \frac{1 - \cos(2\theta)}{2}\\
+&\cos^2\theta = \frac{1 + \cos(2\theta)}{2}
+\end{align*}
+* Product-to-sum formulae
+\begin{align*}
+&2 \cos\theta \cos\varphi = \cos(\theta-\varphi) + \cos(\theta+\varphi)\\
+&2 \sin\theta \sin\varphi = \cos(\theta-\varphi) - \cos(\theta+\varphi)\\
+&2 \sin\theta \cos\varphi = \sin(\theta+\varphi) + \sin(\theta-\varphi)\\
+&2\cos\theta\sin\varphi = \sin(\theta+\varphi) - \sin(\theta-\varphi)
+\end{align*}
+* Sum-to-product formulae
+\begin{align*}
+&\sin\theta \pm \sin\varphi = 2 \sin\left(\frac{\theta\pm\varphi}{2}\right) \cos\left(\frac{\theta\mp\varphi}{2}\right)\\
+&\cos\theta + \cos\varphi = 2 \cos\left(\frac{\theta+\varphi}{2}\right) \cos\left(\frac{\theta-\varphi}{2}\right)\\
+&\tan\theta \pm \tan\varphi = \frac{\sin(\theta\pm\varphi)}{\cos\theta \cos\varphi}\\
+&\cos\theta - \cos\varphi = - 2 \sin\left(\frac{\theta+\varphi}{2}\right) \sin\left(\frac{\theta-\varphi}{2}\right)
+\end{align*}
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