Trigonometry


Some useful references on Trigonometry:

Basic Trigonometric Identities:

$$\begin{aligned}\sin^{2}{\theta} + \cos^{2} \theta &= 1 \\ 1 + \tan^{2} \theta &= \sec^{2} \theta \\ 1 + \cot^{2} \theta &= \csc^{2} \theta \end{aligned}$$

Compound Angle Relations:

$$\begin{aligned} \sin{\left( A \pm B \right)} &= \sin{A}\cos{B} \pm \cos{A}\sin{B} \\ \cos{\left( A \pm B \right)} &= \cos{A}\cos{B} \mp \sin{A}\sin{B} \\ \tan{\left(A \pm B \right)} &= \frac{\tan{A} \pm \tan{B}}{1 \mp \tan{A}\tan{B}} \end{aligned}$$

Product To Sum Formula:

$$\begin{aligned} \sin{u}\sin{v} &= \frac{1}{2}\left[ \cos{\left(u-v \right)} – \cos{\left(u+v \right)} \right] \\ \cos{u}\cos{v} &= \frac{1}{2}\left[ \cos{\left(u-v \right)} + \cos{\left(u+v \right)} \right] \\ \sin{u}\cos{v} &= \frac{1}{2}\left[ \sin{\left(u+v \right)} + \sin{\left(u-v \right)} \right] \\ \cos{u}\sin{v} &= \frac{1}{2}\left[ \sin{\left(u+v \right)} – \sin{\left(u-v \right)} \right] \end{aligned}$$

Double Angle Formulae:

$$\begin{aligned} \sin{2A} &= 2\sin{A}\cos{A} \\ \\ \cos{2A} &= \cos^{2}{A}-\sin^{2}{A} \\ &= 2 \cos^{2}{A}-1 \\ &= 1-2\sin^{2}{A} \\ \\ \tan{2A} &= \frac{2\tan{A}}{1-\tan^{2}{A}} \end{aligned}$$

Factor Formulae:

$$\begin{aligned} \sin{S}+\sin{T} &= 2\sin{\frac{S+T}{2}}\cos{\frac{S-T}{2}} \\ \sin{S}-\sin{T} &= 2\cos{\frac{S+T}{2}}\sin{\frac{S-T}{2}} \\ \cos{S}+\cos{T} &= 2\cos{\frac{S+T}{2}}\cos{\frac{S-T}{2}} \\\cos{S}-\cos{T} &= -2\sin{\frac{S+T}{2}}\sin{\frac{S-T}{2}} \end{aligned}$$

Some Extra Things:

$$\begin{aligned} \cos{x} &= \frac{e^{ix}+e^{-ix}}{2} \\ \sin{x} &= \frac{e^{ix}-e^{-ix}}{2i} \end{aligned}$$

R-Formula:

$$\begin{aligned} a\cos{\theta} \pm b\sin{\theta} &= R\cos{\left( \theta \mp \alpha \right)} \\ a\sin{\theta} \pm b \cos{\theta} &= R \sin{\left(\theta \pm \alpha \right)} \end{aligned}$$

,where
$R = \sqrt{a^{2}+b^{2}}$ and,
$\tan{\alpha} = \frac{b}{a}$ and,
and a > 0, b > 0, α is acute

Note: 
The maximum value is R which occurs when $\cos / \sin{\left( \theta \pm \alpha \right)} = 1$
The minimum value is -R which occurs when $\cos / \sin{\left( \theta \pm \alpha \right)} = -1$

Even/Odd Functions

$$\begin{aligned}\sin{\left(-\theta \right)} &=  -\sin{\theta} \\ \cos{\left( -\theta \right)} &= \cos{\theta} \\ \tan{\left( -\theta \right)} &= -\tan{\theta} \end{aligned}$$
$$\begin{aligned}\csc{\left(-\theta \right)} &=  -\csc{\theta} \\ \sec{\left( -\theta \right)} &= \sec{\theta} \\ \cot{\left( -\theta \right)} &= -\cot{\theta} \end{aligned}$$

Cofunction Identities

$$\begin{aligned}\sin{\left(\frac{\pi}{2}-\theta \right)} &=  \cos{\theta} \\ \csc{\left( \frac{\pi}{2}-\theta \right)} &= \sec{\theta} \\ \tan{\left( \frac{\pi}{2}-\theta \right)} &= \cot{\theta} \end{aligned}$$
$$\begin{aligned}\cos{\left(\frac{\pi}{2}-\theta \right)} &=  \sin{\theta} \\ \sec{\left( \frac{\pi}{2}-\theta \right)} &= \csc{\theta} \\ \cot{\left( \frac{\pi}{2}-\theta \right)} &= \tan{\theta} \end{aligned}$$

 

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