Calculus/Table of Trigonometry

Definitions

 * $$\tan(x)=\frac{\sin(x)}{\cos(x)}$$
 * $$\sec(x)=\frac{1}{\cos(x)}$$
 * $$\cot(x)=\frac{\cos(x)}{\sin(x)}=\frac{1}{\tan(x)}$$
 * $$\csc(x)=\frac{1}{\sin(x)}$$

Inverse trigonometric functions

 * $$\arcsin(x) = \int_0^x \frac{1}{\sqrt{1-t^2}}\mathrm{d}t= -i\log(ix + \sqrt{1-x^2})$$
 * $$\arccos(x) = \frac{\pi}{2} - \arcsin(x) = \frac{\pi}{2} - \int_0^x \frac{1}{\sqrt{1-t^2}}\mathrm{d}t = \frac{\pi}{2} + i\log(ix + \sqrt{1-x^2})$$
 * $$\arctan(x) = \int_0^x \frac{1}{1+t^2} \mathrm{d}t = \frac{i}{2}\log\left(\frac{1-ix}{1+ix}\right)$$
 * $$\arccsc(x) = \arcsin\left(\frac{1}{x}\right) = -i\log\left(\frac{i}{x} + \sqrt{1 - \frac{1}{z^2}}\right)$$
 * $$\arcsec(x) = \arccos\left(\frac{1}{x}\right) = \frac{\pi}{2} - \arcsin\left(\frac{1}{x}\right) = \frac{\pi}{2} + i\log\left(\frac{i}{x} + \sqrt{1 - \frac{1}{z^2}}\right)$$
 * $$\arccot(x) = \arctan\left(\frac{1}{x}\right) = \frac{\pi}{2} - \arctan(x) = \frac{\pi}{2} + \frac{i}{2}\log\left(\frac{1+ix}{1-ix}\right)$$
 * $$\arcsin(x) + \arcsin(y) = \arcsin\left(x\sqrt{1-y^2} + y\sqrt{1-x^2}\right)$$
 * $$\arccos(x) + \arccos(y) = \arccos\left(xy - \sqrt{(1-x^2)(1-y^2)}\right)$$
 * $$\arctan(x) + \arctan(y) = \arctan\left(\frac{x+y}{1-xy}\right) \pmod \pi$$

Pythagorean Identities

 * $$\sin^2(x)+\cos^2(x)=1$$
 * $$1+\tan^2(x)=\sec^2(x)$$
 * $$1+\cot^2(x)=\csc^2(x)$$

Double Angle Identities

 * $$\sin(2x)=2\sin(x)\cos(x)$$
 * $$\cos(2x)=\cos^2(x)-\sin^2(x)$$
 * $$\tan(2x)=\frac{2\tan(x)}{1-\tan^2(x)}$$
 * $$\cos^2(x)=\frac{1+\cos(2x)}{2}$$
 * $$\sin^2(x)=\frac{1-\cos(2x)}{2}$$

Angle Sum Identities

 * $$\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)$$


 * $$\sin(x-y)=\sin(x)\cos(y)-\cos(x)\sin(y)$$


 * $$\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)$$


 * $$\cos(x-y)=\cos(x)\cos(y)+\sin(x)\sin(y)$$


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


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


 * $$\cos(x)+\cos(y)=2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)$$


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


 * $$\tan(x)+\tan(y)=\frac{\sin(x+y)}{\cos(x)\cos(y)}$$


 * $$\tan(x)-\tan(y)=\frac{\sin(x-y)}{\cos(x)\cos(y)}$$


 * $$\cot(x)+\cot(y)=\frac{\sin(x+y)}{\sin(x)\sin(y)}$$


 * $$\cot(x)-\cot(y)=\frac{-\sin(x-y)}{\sin(x)\sin(y)}$$

Product-to-sum identities

 * $$\cos(x)\cos(y)= \frac{\cos(x+y)+\cos(x-y)}{2}$$


 * $$\sin(x)\sin(y)=\frac{\cos(x-y)-\cos(x+y)}{2}$$


 * $$\sin(x)\cos(y)=\frac{\sin(x+y)+\sin(x-y)}{2}$$


 * $$\cos(x)\sin(y)=\frac{\sin(x+y)-\sin(x-y)}{2}$$

In terms of the complex exponential

 * $$e^{i\theta} = \mathrm{cis} \theta = i\sin \theta + \cos\theta$$
 * $$\sin\theta = \mathrm{Re}(e^{i\theta}) = \frac{e^{i\theta}-e^{-i\theta}}{2i}$$
 * $$\cos\theta = \mathrm{Im}(e^{i\theta}) = \frac{e^{i\theta}+e^{-i\theta}}{2}$$
 * $$\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{e^{2i\theta}-1}{i(e^{2i\theta}+1)}$$
 * $$\csc\theta = \frac{1}{\sin\theta} = \frac{2i}{e^{i\theta}-e^{-i\theta}}$$
 * $$\sec\theta = \frac{1}{\cos\theta} = \frac{2}{e^{i\theta}+e^{-i\theta}}$$
 * $$\cot\theta = \frac{1}{\tan\theta} = \frac{i(e^{2i\theta}+1)}{e^{2i\theta}-1}$$

Hyperbolic functions

 * $$e^x = \sinh x + \cosh x$$
 * $$\cosh^2 x - \sinh^2 x = 1$$
 * $$\mathrm{sech}^2 x = 1 - \tanh^2 x$$
 * $$\mathrm{csch}^2 x = \mathrm{coth}^2 x - 1$$
 * $$\sinh x = -i\sin ix = \frac{e^{x}-e^{-x}}{2}$$
 * $$\cosh x = \cos ix = \frac{e^{x}+e^{-x}}{2}$$
 * $$\tanh x = -i\tan ix = \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$$
 * $$\mathrm{csch} x = i\csc ix = \frac{2}{e^{x}-e^{-x}}$$
 * $$\mathrm{sech} x = \sec ix = \frac{2}{e^{x}+e^{-x}}$$
 * $$\mathrm{coth} x = i\cot ix = \frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}$$

Inverses

 * $$\mathrm{arsinh} x = \int_0^x \frac{1}{\sqrt{t^2 + 1}} \mathrm{d}t = \log\left(x + \sqrt{x^2 + 1}\right)$$
 * $$\mathrm{arcosh} x = \int_1^x \frac{1}{\sqrt{t^2 - 1}} \mathrm{d}t = \log\left(x + \sqrt{x^2 - 1}\right)$$
 * $$\mathrm{artanh} x = \int_0^x \frac{1}{1-t^2} \mathrm{d}t = \frac{1}{2}\log\left(\frac{1+x}{1-x}\right)$$
 * $$\mathrm{arccsh} x = \log\left(\frac{1 + \sqrt{1 + x^2}}{x}\right)$$
 * $$\mathrm{arsech} x = \log\left(\frac{1 + \sqrt{1 - x^2}}{x}\right)$$
 * $$\mathrm{arcoth} x = \frac{1}{2}\log\left(\frac{x+1}{x-1}\right)$$