Fundamental trigonometric functions

Relation of right triangle sides are defined as trigonometric functions. They are:

Trignometric Function	Definition	Mathematical Formula	Graphs
Sine	Ratio of opposite side over hypotnuse	${\displaystyle \sin A={\frac {a}{h}}}$
Cosine	Ratio of adjacent side over hypotnuse	${\displaystyle \cos A={\frac {b}{h}}}$
Tangent	Ratio of opposite over Adjacent	${\displaystyle \tan A={\frac {a}{b}}}$
Co tangent	Ratio of adjacent over opposite	${\displaystyle \cot A={\frac {b}{a}}}$
Secant	Ratio of 1 over opposite	${\displaystyle \sec A={\frac {h}{b}}}$
Co secant	Ratio of 1 over adjacent	${\displaystyle \csc A={\frac {h}{a}}}$

Chracteristics

Periodic	${\displaystyle \sin(x)=\sin(x+2k\pi )\,}$ ${\displaystyle \cos(x)=\cos(x+2k\pi )\,}$ ${\displaystyle \tan(x)=\tan(x+k\pi )\,}$
Symmetry	${\displaystyle \sin(-x)=-\sin(x)\,}$ ${\displaystyle \cos(-x)=\;\cos(x)\,}$ ${\displaystyle \tan(-x)=-\tan(x)\,}$
Phase shift	${\displaystyle \sin(x)=\cos \left({\frac {\pi }{2}}-x\right)}$ ${\displaystyle \cos(x)=\sin \left({\frac {\pi }{2}}-x\right)}$ ${\displaystyle \tan(x)=\cot \left({\frac {\pi }{2}}-x\right)}$
Complex Power	${\displaystyle \cos(x)={\frac {e^{ix}+e^{-ix}}{2}}\;}$ ${\displaystyle \sin(x)={\frac {e^{ix}-e^{-ix}}{2i}}\;}$ với ${\displaystyle i^{2}=-1.\,}$ ${\displaystyle e^{\imath x}=\cos(x)+\imath \sin(x)\,}$ và ${\displaystyle \imath ={\sqrt {-1}}.\,}$ de Moivre's formula: ${\displaystyle \cos(nx)+i\sin(nx)=(\cos(x)+i\sin(x))^{n}\,}$
Limit	${\displaystyle \lim _{x\rightarrow 0}{\frac {\sin(x)}{x}}=1,}$ ${\displaystyle \lim _{x\rightarrow 0}{\frac {1-\cos(x)}{x}}=0,}$
Derivative	${\displaystyle {d \over dx}\cos(x)=-\sin(x)}$ ${\displaystyle {d \over dx}\sin(x)=\cos(x)}$

Fundamental trigonometry function

${\displaystyle \tan(x)={\frac {\sin(x)}{\cos(x)}}\qquad \operatorname {cotg} (x)={\frac {\cos(x)}{\sin(x)}}={\frac {1}{\tan(x)}}}$

Sum and difference angles

${\displaystyle \sin(x\pm y)=\sin(x)\cos(y)\pm \cos(x)\sin(y)\,}$

${\displaystyle \cos(x\pm y)=\cos(x)\cos(y)\mp \sin(x)\sin(y)\,}$

${\displaystyle \tan(x\pm y)={\frac {\tan(x)\pm \tan(y)}{1\mp \tan(x)\tan(y)}}}$

Reduced power

${\displaystyle \cos ^{2}(x)={1+\cos(2x) \over 2}}$

${\displaystyle \sin ^{2}(x)={1-\cos(2x) \over 2}}$

${\displaystyle \sin ^{2}(x)\cos ^{2}2(x)={1-\cos(4x) \over 4}}$

${\displaystyle \sin ^{3}(x)={\frac {23\sin 2(x)-\sin(3x)}{4}}}$

${\displaystyle \cos ^{3}(x)={\frac {32\cos(x)+\cos(3x)}{4}}}$

Half angles

${\displaystyle \cos \left({\frac {x}{2}}\right)=\pm \,{\sqrt {\frac {1+\cos(x)}{2}}}}$

${\displaystyle \sin \left({\frac {x}{2}}\right)=\pm \,{\sqrt {\frac {1-\cos(x)}{2}}}}$

${\displaystyle \tan \left({\frac {x}{2}}\right)={\sin(x/2) \over \cos(x/2)}=\pm \,{\sqrt {1-\cos x \over 1+\cos x}}.\qquad \qquad (1)}$

${\displaystyle \tan \left({\frac {x}{2}}\right)=\pm \,{\sqrt {(1-\cos x)(1+\cos x) \over (1+\cos x)(1+\cos x)}}=\pm \,{\sqrt {1-\cos ^{2}x \over (1+\cos x)^{2}}}}$

${\displaystyle ={\sin x \over 1+\cos x}.}$

${\displaystyle \tan \left({\frac {x}{2}}\right)=\pm \,{\sqrt {(1-\cos x)(1-\cos x) \over (1+\cos x)(1-\cos x)}}=\pm \,{\sqrt {(1-\cos x)^{2} \over (1-\cos ^{2}x)}}}$

${\displaystyle ={1-\cos x \over \sin x}.}$

${\displaystyle \tan \left({\frac {x}{2}}\right)={\frac {\sin(x)}{1+\cos(x)}}={\frac {1-\cos(x)}{\sin(x)}}.}$

${\displaystyle t=\tan \left({\frac {x}{2}}\right),}$

${\displaystyle \sin(x)={\frac {2t}{1+t^{2}}}}$

and

${\displaystyle \cos(x)={\frac {1-t^{2}}{1+t^{2}}}}$

and

${\displaystyle e^{ix}={\frac {1+it}{1-it}}.}$

Product of trigonomtric functions

${\displaystyle \cos \left(x\right)\cos \left(y\right)={\cos \left(x+y\right)+\cos \left(x-y\right) \over 2}\;}$

${\displaystyle \sin \left(x\right)\sin \left(y\right)={\cos \left(x-y\right)-\cos \left(x+y\right) \over 2}\;}$

${\displaystyle \sin \left(x\right)\cos \left(y\right)={\sin \left(x-y\right)+\sin \left(x+y\right) \over 2}\;}$

Inverse trigonometric functions

${\displaystyle \arcsin(x)+\arccos(x)=\pi /2\;}$

${\displaystyle \arctan(x)+\operatorname {arccot}(x)=\pi /2.\;}$

${\displaystyle \arctan(x)+\arctan(1/x)=\left\{{\begin{matrix}\pi /2,&{\mbox{n}}{\acute {\hat {\mbox{e}}}}{\mbox{u}}\ x>0\\-\pi /2,&{\mbox{n}}{\acute {\hat {\mbox{e}}}}{\mbox{u}}\ x<0\end{matrix}}\right..}$

${\displaystyle \arctan(x)+\arctan(y)=\arctan \left({\frac {x+y}{1-xy}}\right)\;}$

${\displaystyle \arctan(x)-\arctan(y)=\arctan \left({\frac {x-y}{1+xy}}\right)\;}$

${\displaystyle \sin(\arccos(x))={\sqrt {1-x^{2}}}\,}$

${\displaystyle \cos(\arcsin(x))={\sqrt {1-x^{2}}}\,}$

${\displaystyle \sin(\arctan(x))={\frac {x}{\sqrt {1+x^{2}}}}}$

${\displaystyle \cos(\arctan(x))={\frac {1}{\sqrt {1+x^{2}}}}}$

${\displaystyle \tan(\arcsin(x))={\frac {x}{\sqrt {1-x^{2}}}}}$

${\displaystyle \tan(\arccos(x))={\frac {\sqrt {1-x^{2}}}{x}}}$

Complex number

${\displaystyle \cos(x)={\frac {e^{ix}+e^{-ix}}{2}}\;}$

${\displaystyle \sin(x)={\frac {e^{ix}-e^{-ix}}{2i}}\;}$

${\displaystyle i^{2}=-1.\,}$

With

${\displaystyle e^{\imath x}=\cos(x)+\imath \sin(x)\,}$

Hence

${\displaystyle \imath ={\sqrt {-1}}.\,}$

Infinite Product

${\displaystyle \sin x=x\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}n^{2}}}\right)}$

${\displaystyle \sinh x=x\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}n^{2}}}\right)}$

${\displaystyle \cos x=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}(n-{\frac {1}{2}})^{2}}}\right)}$

${\displaystyle \cosh x=\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}(n-{\frac {1}{2}})^{2}}}\right)}$

${\displaystyle {\frac {\sin x}{x}}=\prod _{n=1}^{\infty }\cos \left({\frac {x}{2^{n}}}\right)}$

Limits

${\displaystyle \lim _{x\rightarrow 0}{\frac {\sin(x)}{x}}=1,}$

${\displaystyle \lim _{x\rightarrow 0}{\frac {1-\cos(x)}{x}}=0,}$

Derivatives

${\displaystyle {d \over dx}\cos(x)=-\sin(x)}$

${\displaystyle {d \over dx}\sin(x)=\cos(x)}$

${\displaystyle {d \over dx}\tan(x)=\sec ^{2}(x)}$

${\displaystyle {d \over dx}\cot(x)=-\csc ^{2}(x)}$

${\displaystyle {d \over dx}\sec(x)=\sec(x)\tan(x)}$

${\displaystyle {d \over dx}\csc(x)=-\csc(x)\cot(x)}$

${\displaystyle {d \over dx}\arcsin(x)={\frac {1}{\sqrt {1-x^{2}}}}}$

${\displaystyle {d \over dx}\arctan(x)={\frac {1}{1+x^{2}}}}$

Common Trigonomtric functions

${\displaystyle \sin \left(x+y\right)=\sin x\cos y+\cos x\sin y}$

${\displaystyle \sin \left(x-y\right)=\sin x\cos y-\cos x\sin y}$

${\displaystyle \cos \left(x+y\right)=\cos x\cos y-\sin x\sin y}$

${\displaystyle \cos \left(x-y\right)=\cos x\cos y+\sin x\sin y}$

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

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

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

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

${\displaystyle \tan x+\tan y={\frac {\sin \left(x+y\right)}{\cos x\cos y}}}$

${\displaystyle \tan x-\tan y={\frac {\sin \left(x-y\right)}{\cos x\cos y}}}$

${\displaystyle \cot x+\cot y={\frac {\sin \left(x+y\right)}{\sin x\sin y}}}$

${\displaystyle \cot x-\cot y={\frac {-\sin \left(x-y\right)}{\sin x\sin y}}}$

Double angle trigonometric functions

${\displaystyle \sin(2x)=2\sin(x)\cos(x)\,}$

${\displaystyle \cos(2x)=\cos ^{2}(x)-\sin ^{2}(x)=2\cos ^{2}(x)-1=1-2\sin ^{2}(x)\,}$

${\displaystyle \tan(2x)={\frac {2\tan(x)}{1-\tan ^{2}(x)}}}$

In general,

If T_n is Chebyshev n ordered then

${\displaystyle \cos(nx)=T_{n}(\cos(x)).\,}$

De Moivre formula:

${\displaystyle \cos(nx)+i\sin(nx)=(\cos(x)+i\sin(x))^{n}\,}$

hạt nhân Dirichlet D_n(x) sẽ xuất hiện trong các công thức sau:

${\displaystyle 1+2\cos(x)+2\cos(2x)+2\cos(3x)+\cdots +2\cos(nx)\;}$

${\displaystyle ={\frac {\sin \left(\left(n+{\frac {1}{2}}\right)x\right)}{\sin(x/2)}}\;}$

${\displaystyle \sin(nx)=2\sin((n-1)x)\cos(x)-\sin((n-2)x)}$

${\displaystyle \cos(nx)=2\cos((n-1)x)\cos(x)-\cos((n-2)x)}$

Triple angle trigonometric functions

${\displaystyle \sin(3x)=3\sin(x)-4\sin ^{3}(x)}$

${\displaystyle \cos(3x)=4\cos ^{3}(x)-3\cos(x)}$

Series

${\displaystyle {\begin{matrix}\arcsin z&=&z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{7}}{7}}+\cdots \\&=&\sum _{n=0}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {z^{2n+1}}{(2n+1)}}\end{matrix}}\,\quad \left|z\right|<1}$

${\displaystyle {\begin{matrix}\arccos z&=&{\frac {\pi }{2}}-\arcsin z\\&=&{\frac {\pi }{2}}-(z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{7}}{7}}+\cdots )\\&=&{\frac {\pi }{2}}-\sum _{n=0}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {z^{2n+1}}{(2n+1)}}\end{matrix}}\,\quad \left|z\right|<1}$

${\displaystyle {\begin{matrix}\arctan z&=&z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots \\&=&\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}}\end{matrix}}\,\quad \left|z\right|<1}$

${\displaystyle {\begin{matrix}\operatorname {arccsc} z&=&\arcsin \left(z^{-1}\right)\\&=&z^{-1}+\left({\frac {1}{2}}\right){\frac {z^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{-5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{-7}}{7}}+\cdots \\&=&\sum _{n=0}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {z^{-(2n+1)}}{2n+1}}\end{matrix}}\,\quad \left|z\right|>1}$

${\displaystyle {\begin{matrix}\operatorname {arcsec} z&=&\arccos \left(z^{-1}\right)\\&=&{\frac {\pi }{2}}-(z^{-1}+\left({\frac {1}{2}}\right){\frac {z^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{-5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{-7}}{7}}+\cdots )\\&=&{\frac {\pi }{2}}-\sum _{n=0}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {z^{-(2n+1)}}{(2n+1)}}\end{matrix}}\,\quad \left|z\right|>1}$

${\displaystyle {\begin{matrix}\operatorname {arccot} z&=&{\frac {\pi }{2}}-\arctan z\\&=&{\frac {\pi }{2}}-(z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots )\\&=&{\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}}\end{matrix}}\,\quad \left|z\right|<1}$

Integrals

${\displaystyle \arcsin \left(x\right)=\int _{0}^{x}{\frac {1}{\sqrt {1-z^{2}}}}\,\mathrm {d} z,\quad |x|<1}$

${\displaystyle \arccos \left(x\right)=\int _{x}^{1}{\frac {1}{\sqrt {1-z^{2}}}}\,\mathrm {d} z,\quad |x|<1}$

${\displaystyle \arctan \left(x\right)=\int _{0}^{x}{\frac {1}{1+z^{2}}}\,\mathrm {d} z,\quad \forall x\in \mathbb {R} }$

${\displaystyle \operatorname {arccot} \left(x\right)=\int _{x}^{\infty }{\frac {1}{z^{2}+1}}\,\mathrm {d} z,\quad z>0}$

${\displaystyle \operatorname {arcsec} \left(x\right)=\int _{x}^{1}{\frac {1}{|z|{\sqrt {z^{2}-1}}}}\,\mathrm {d} z,\quad x>1}$

${\displaystyle \operatorname {arccsc} \left(x\right)=\int _{x}^{\infty }{\frac {-1}{|z|{\sqrt {z^{2}-1}}}}\,\mathrm {d} z,\quad x>1}$

Complex trigonometric functions

${\displaystyle \arcsin(z)=-i\log \left(i\left(z+{\sqrt {1-z^{2}}}\right)\right)}$

${\displaystyle \arccos(z)=-i\log \left(z+{\sqrt {z^{2}-1}}\right)}$

${\displaystyle \arctan(z)={\frac {i}{2}}\log \left({\frac {1-iz}{1+iz}}\right)}$