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Maths formulas

Trigonometry


\[S = \frac{O}{H}\] \[C = \frac{A}{H}\] \[T = \frac{O}{A}\]
\[\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}\] \[\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}\] \[\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}\]
\[\tan\theta = \frac{\sin\theta}{\cos\theta}\] \[\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}\] \[\sec\theta = \frac{1}{\cos\theta}\] \[\csc\theta = \frac{1}{\sin\theta}\]
\[\cos^2 \theta + \sin^2 \theta = 1\]
\[\sin \left(\frac{\pi}{2} - \theta \right) = \cos\theta \] \[\cos \left(\frac{\pi}{2} - \theta \right) = \sin\theta\]
\[\cot \left(\frac{\pi}{2} - \theta \right) = \tan\theta\] \[\tan \left(\frac{\pi}{2} - \theta \right) = \cot\theta\]
\[\csc \left(\frac{\pi}{2} - \theta \right) = \sec\theta\] \[\sec \left(\frac{\pi}{2} - \theta \right) = \csc\theta\]
\[\sin (\pi - \theta) = \sin \theta\] \[\cos (\pi - \theta) = - \cos \theta\] \[\tan (\pi - \theta) = - \tan \theta\]
Addition formulas are also known as Ptolemy’s identities
\[\sin (\alpha+\beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\] \[\sin (\alpha-\beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta\]
\[\cos (\alpha+\beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\] \[\cos (\alpha-\beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\]
\[\tan (\alpha+\beta) = \frac{\tan \alpha + \tan \beta}{1- \tan \alpha \tan \beta}\] \[\tan (\alpha-\beta) = \frac{\tan \alpha - \tan \beta}{1+ \tan \alpha \tan \beta}\]
Double angle formulas
\[\sin 2\theta = 2 \sin \theta \cos \theta\] \[\begin{eqnarray*}\cos 2\theta &=& \cos^2 \theta - \sin^2 \theta \\ &=& 1 - 2 \sin^2 \theta \\ &=& 2 \cos^2 \theta - 1\end{eqnarray*}\] \[\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}\]
Sum to product (Factorisation)
\[\sin \alpha + \sin \beta = 2 \sin \frac{\alpha + \beta}{2} \cos \frac{\alpha - \beta}{2}\] \[\sin \alpha - \sin \beta = 2 \sin \frac{\alpha - \beta}{2} \cos \frac{\alpha + \beta}{2}\]
\[\cos \alpha + \cos \beta = 2 \cos \frac{\alpha + \beta}{2} \cos \frac{\alpha - \beta}{2}\] \[\cos \alpha - \cos \beta = -2 \sin \frac{\alpha + \beta}{2} \sin \frac{\alpha - \beta}{2}\]
Product to sum
\[\sin \alpha \cos \beta = \frac{1}{2} \left[ \sin (\alpha + \beta) + \sin (\alpha - \beta) \right]\] \[\cos \alpha \cos \beta = \frac{1}{2} \left[ \cos (\alpha + \beta) + \cos (\alpha - \beta) \right]\]
\[\begin{eqnarray*} \sin \alpha \sin \beta &=& -\frac{1}{2} \left[ \cos (\alpha + \beta) - \cos (\alpha - \beta) \right] \\ &=& \frac{1}{2} \left[ \cos (\alpha - \beta) - \cos (\alpha + \beta) \right]\end{eqnarray*}\]
Linearisation
\[\sin^2 \theta = \frac{1}{2} - \frac{1}{2} \cos 2 \theta\] \[\cos^2 \theta = \frac{1}{2} + \frac{1}{2} \cos 2 \theta\]
\[\sin^3 \theta = \frac{3}{4} \sin \theta - \frac{1}{4} \sin 3 \theta\] \[\cos^3 \theta = \frac{3}{4} \cos \theta + \frac{1}{4} \cos 3 \theta\]

\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\] or \[\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\]
\[\text{Area} = \frac{1}{2} ab \sin C\] or \[\text{Area} = \frac{1}{2} ac \sin B\] or \[\text{Area} = \frac{1}{2} bc \sin A\]

\[c^2 = a^2 + b^2 - 2ab \cos C\] \[b^2 = a^2 + c^2 - 2ac \cos B\] \[a^2 = b^2 + c^2 - 2bc \cos A\]
\[\cos C = \frac{a^2 + b^2 - c^2}{2ab}\] \[\cos B = \frac{a^2 + c^2 - b^2}{2ac}\] \[\cos A = \frac{b^2 + c^2 - a^2}{2bc}\]