# Trigonometry formulas

## Right-Triangle Definitions

$$\sin \alpha = \frac{\text{Opposite}}{\text{Hypotenuse}}$$

$$\cos \alpha = \frac{\text{Adjacent}}{\text{Hypotenuse}}$$

$$\tan \alpha = \frac{\text{Opposite}}{\text{Adjacent}}$$

$$\csc \alpha = \frac{1}{\sin\alpha} = \frac{\text{Hypotenuse}}{\text{Opposite}}$$

$$\sec \alpha = \frac{1}{\cos\alpha} = \frac{\text{Hypotenuse}}{\text{Adjacent}}$$

$$\cot \alpha = \frac{1}{\tan\alpha} = \frac{\text{Adjacent}}{\text{Opposite}}$$

## 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

$$A = \frac{1}{2} a\,b\, \sin\gamma$$