angular-methods-list.md

Angular Methods

Function	$f(x)$	$f^{\prime }(x)$	$f^{-1}(x)$
sine	$\sin (x)$	$\cos (x)$	$\arcsin (x)$
cosine	$\cos (x)$	$-\sin (x)$	$\arccos (x)$
tangent	$\tan (x)$	${\sec }^2(x)=\frac{1}{{\cos }^2(x)}$	$\arctan (x)$
cosecant	$\csc (x)=\frac{1}{\sin (x)}$	$-\csc (x)\cot (x)=\frac{-1}{\sin (x)\tan (x)}$	$\arcsin \left(\frac{1}{x}\right)$
secant	$\sec (x)=\frac{1}{\cos (x)}$	$\sec (x)\tan (x)=\frac{\tan (x)}{\cos (x)}$	$\arccos \left(\frac{1}{x}\right)$
cotangent	$\cot (x)=\frac{1}{\tan (x)}$	$-{\csc }^2(x)=\frac{-1}{{\sin }^2(x)}$	$\arctan \left(\frac{1}{x}\right)$
exsec[ant] (external secant)	$\text{exs}(x)=\frac{1}{\cos (x)}-1$	$\tan (x)\sec (x)=\frac{\sin (x)}{{\cos }^2(x)}$	$\text{arcsec}(x+1)=\arccos \left(\frac{1}{x+1}\right)=\arctan (\sqrt{x^2+2x})$
excosec[ant]/coexsec[ant] (external cosecant)	$\text{exc}(x)=\frac{1}{\sin (x)}-1$	$-\cot (x)\csc (x)=\frac{-\cos (x)}{{\sin }^2(x)}$	$\text{arccsc}(x+1)=\arcsin \left(\frac{1}{x+1}\right)$
versin (versed sine)	$\text{ver}(x)=2{\sin }^2\left(\frac{x}{2}\right)=1-\cos (x)$	$\sin (x)$	$\arccos (1-x)$
coversin (coversed sine)	$\text{cvs}(x)=1-\sin (x)$	$-\cos (x)$	$\arcsin (1-x)$
vercos (versed cosine)	$\text{vcs}(x)=2{\cos }^2\left(\frac{x}{2}\right)=1+\cos (x)$	$-\sin (x)$	$\arccos (x-1)$
covercos (coversed cosine)	$\text{cvc}(x)=1+\sin (x)$	$\cos (x)$	$\arcsin (x-1)$
haversin (half-versed sine)	$\text{hav}(x)={\sin }^2\left(\frac{x}{2}\right)=\frac{1-\cos (x)}{2}$	$\frac{\sin (x)}{2}$	$2\arcsin (\sqrt{x})=\arccos (1-2x)$
hacoversin (half-coversed sine)	$\text{hcv}(x)=\frac{1-\sin (x)}{2}$	$\frac{-\cos (x)}{2}$	$\arcsin (1-2x)$
havercos (half-versed cosine)	$\text{hvc}(x)={\cos }^2\left(\frac{x}{2}\right)=\frac{1+\cos (x)}{2}$	$\frac{-\sin (x)}{2}$	$2\arccos (\sqrt{x})=\arccos (2x-1)$
hacovercos (half-coversed cosine)	$\text{hcc}(x)=\frac{1+\sin (x)}{2}$	$\frac{\cos (x)}{2}$	$\arcsin (2x-1)$

In terms of naming, *sin = *sine, as well as *cos = *cosin(e).

for $x\in (0;\pi )$: ${\csc }^{\prime }=-\text{arcsinh}(\cot (x))$ for $x\in (-\frac{\pi }{2};\frac{\pi }{2})$: ${\sec }^{\prime }(x)=\text{arcsinh}(\tan (x))$