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

• $\arcsin (x)=\underset{0}{\overset{x}{\int }}\frac{1}{\sqrt{1-t^2}}\mathrm{d}t=-i\log (ix+\sqrt{1-x^2})$
• $\arccos (x)=\frac{\pi }{2}-\arcsin (x)=\frac{\pi }{2}-\underset{0}{\overset{x}{\int }}\frac{1}{\sqrt{1-t^2}}\mathrm{d}t=\frac{\pi }{2}+i\log (ix+\sqrt{1-x^2})$
• $\arctan (x)=\underset{0}{\overset{x}{\int }}\frac{1}{1+t^2}\mathrm{d}t=\frac{i}{2}\log \left(\frac{1-ix}{1+ix}\right)$
• $\mathrm{arccsc}(x)=\arcsin \left(\frac{1}{x}\right)=-i\log (\frac{i}{x}+\sqrt{1-\frac{1}{z^2}})$
• $\mathrm{arcsec}(x)=\arccos \left(\frac{1}{x}\right)=\frac{\pi }{2}-\arcsin \left(\frac{1}{x}\right)=\frac{\pi }{2}+i\log (\frac{i}{x}+\sqrt{1-\frac{1}{z^2}})$
• $\mathrm{arccot}(x)=\arctan \left(\frac{1}{x}\right)=\frac{\pi }{2}-\arctan (x)=\frac{\pi }{2}+\frac{i}{2}\log \left(\frac{1+ix}{1-ix}\right)$
• $\arcsin (x)+\arcsin (y)=\arcsin (x\sqrt{1-y^2}+y\sqrt{1-x^2})$
• $\arccos (x)+\arccos (y)=\arccos (xy-\sqrt{(1-x^2)(1-y^2)})$
• $\arctan (x)+\arctan (y)=\arctan \left(\frac{x+y}{1-xy}\right)(\mathrm{mod}\pi )$

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

$e^{i\theta }=\mathrm{c}\mathrm{i}\mathrm{s}\theta =i\sin \theta +\cos \theta$

$\sin \theta =\mathrm{R}\mathrm{e}(e^{i\theta })=\frac{e^{i\theta }-e^{-i\theta }}{2i}$

$\cos \theta =\mathrm{I}\mathrm{m}(e^{i\theta })=\frac{e^{i\theta }+e^{-i\theta }}{2}$

$\tan \theta =\frac{\sin \theta }{\cos \theta }=\frac{e^{2i\theta }-1}{i(e^{2i\theta }+1)}$

$\csc \theta =\frac{1}{\sin \theta }=\frac{2i}{e^{i\theta }-e^{-i\theta }}$

$\sec \theta =\frac{1}{\cos \theta }=\frac{2}{e^{i\theta }+e^{-i\theta }}$

$\cot \theta =\frac{1}{\tan \theta }=\frac{i(e^{2i\theta }+1)}{e^{2i\theta }-1}$

Hyperbolic functions

$e^x=\sinh x+\cosh x$

${\cosh }^{2}x-{\sinh }^{2}x=1$

${\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^{2}x=1-{\tanh }^{2}x$

${\mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}}^{2}x={\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}}^{2}x-1$

$\sinh x=-i\sin ix=\frac{e^x-e^{-x}}{2}$

$\cosh x=\cos ix=\frac{e^x+e^{-x}}{2}$

$\tanh x=-i\tan ix=\frac{e^x-e^{-x}}{e^x+e^{-x}}$

$\mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}x=i\csc ix=\frac{2}{e^x-e^{-x}}$

$\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}x=\sec ix=\frac{2}{e^x+e^{-x}}$

$\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}x=i\cot ix=\frac{e^x+e^{-x}}{e^x-e^{-x}}$

Inverses

$\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}x=\underset{0}{\overset{x}{\int }}\frac{1}{\sqrt{t^2+1}}\mathrm{d}t=\log (x+\sqrt{x^2+1})$

$\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}x=\underset{1}{\overset{x}{\int }}\frac{1}{\sqrt{t^2-1}}\mathrm{d}t=\log (x+\sqrt{x^2-1})$

$\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}x=\underset{0}{\overset{x}{\int }}\frac{1}{1-t^2}\mathrm{d}t=\frac{1}{2}\log \left(\frac{1+x}{1-x}\right)$

$\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{s}\mathrm{h}x=\log \left(\frac{1+\sqrt{1+x^2}}{x}\right)$

$\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}x=\log \left(\frac{1+\sqrt{1-x^2}}{x}\right)$

$\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}x=\frac{1}{2}\log \left(\frac{x+1}{x-1}\right)$

See also

• Trigonometry

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