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Math Formulas: Trigonometry Identities

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

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