# List of formulas relared to Trigonometric functions

#### Right-Triangle Definitions :

$\sin \alpha = \frac{\text{Opposite}}{\text{Hypotenuse}}=\frac{\text{a}}{\text{c}}$ $\cos \alpha = \frac{\text{Adjacent}}{\text{Hypotenuse}}=\frac{\text{b}}{\text{c}}$ $\tan \alpha = \frac{\text{Opposite}}{\text{Adjacent}}=\frac{\text{a}}{\text{b}}$ $\csc \alpha = \frac{1}{\sin\alpha} = \frac{\text{Hypotenuse}}{\text{Opposite}}=\frac{\text{c}}{\text{a}}$ $\sec \alpha = \frac{1}{\cos\alpha} = \frac{\text{Hypotenuse}}{\text{Adjacent}}=\frac{\text{c}}{\text{b}}$ $\cot \alpha = \frac{1}{\tan\alpha} = \frac{\text{Adjacent}}{\text{Opposite}}=\frac{\text{b}}{\text{a}}$

#### Reduction Formulas :

$\sin(-x) = -\sin(x)$ $\cos(-x) = \cos(x)$ $\sin\left(\frac{\pi}{2} - x\right) = \cos(x)$ $\cos\left(\frac{\pi}{2} - x\right) = \sin(x)$ $\sin\left(\frac{\pi}{2} + x\right) = \cos(x)$ $\cos\left(\frac{\pi}{2} + x\right) = -\sin(x)$ $\sin(\pi - x) = \sin(x)$ $\cos(\pi - x) = -\cos(x)$ $\sin(\pi + x) = -\sin(x)$ $\cos(\pi + x) = -\cos(x)$

#### Basic Identities :

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

#### Sum and Difference Formulas :

$\sin(\alpha + \beta) = \sin\alpha \cdot \cos \beta + \sin\beta \cdot \cos\alpha$ $\sin(\alpha - \beta) = \sin\alpha \cdot \cos \beta - \sin \beta \cdot \cos\alpha$ $\cos(\alpha + \beta) = \cos\alpha \cdot \cos \beta - \sin\alpha \cdot \cos\beta$ $\cos(\alpha - \beta) = \cos\alpha \cdot \cos \beta + \sin\alpha \cdot \cos\beta$ $\tan(\alpha + \beta) = \frac{ \tan\alpha + \tan\beta}{1 - \tan\alpha \cdot \tan\beta }$ $\tan(\alpha - \beta) = \frac{ \tan\alpha - \tan\beta}{1 + \tan\alpha \cdot \tan\beta }$

#### Double Angle and Half Angle Formulas :

$\sin(2\,\alpha) = 2 \cdot \sin\alpha \cdot \cos\alpha$ $\cos(2\,\alpha) = \cos^2\alpha - \sin^2\alpha$ $\tan(2\,\alpha) = \frac{2\,\tan\alpha}{1 - \tan^2\alpha}$ $\sin \frac{\alpha}{2} = \pm \sqrt{\frac{1-\cos\alpha}{2}}$ $\cos \frac{\alpha}{2} = \pm \sqrt{\frac{1+\cos\alpha}{2}}$ $\tan \frac{\alpha}{2} = \frac{1 - \cos\alpha}{\sin\alpha} = \frac{\sin\alpha}{1 - \cos\alpha}$ $\tan \frac{\alpha}{2} = \pm \sqrt{\frac{1 + \cos\alpha}{1 - \cos\alpha} }$

#### Other Useful Trig Formulas :

Law of sines $\frac{\sin\alpha}{\alpha} = \frac{\sin\beta}{\beta} = \frac{\sin\gamma}{\gamma}$ Law of cosines \begin{aligned} a^2 = b^2 + c^2 - 2\cdot b\cdot c\cdot \cos\alpha \\ b^2 = a^2 + c^2 - 2\cdot a\cdot c\cdot \cos\beta \\ c^2 = a^2 + b^2 - 2\cdot a\cdot b\cdot \cos\gamma \end{aligned} Area of triangle $A = \frac{1}{2} a\,b\, \sin\gamma$