Arkus sinus

Funkce ${\displaystyle y=\arcsin x}$ je inverzní k funkci ${\displaystyle x=\sin y\;\left(-{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}}\right)}$; je definována pro ${\displaystyle x\in \langle -1,1\rangle }$.[1]

${\displaystyle {\begin{array}{lcll}\arcsin x+\arccos x&=&{\frac {\pi }{2}}\\\arcsin x+\arcsin(-x)&=&0\\\arcsin x+\arcsin y&=&\left\{{\begin{array}{rl}\arcsin \left(x\,{\sqrt {1-y^{2}}}+y\,{\sqrt {1-x^{2}}}\right),&xy\leq 0\;\;{\mbox{nebo}}\;\;x^{2}+y^{2}\leq 1\\\pi -\arcsin \left(x\,{\sqrt {1-y^{2}}}+y\,{\sqrt {1-x^{2}}}\right),&x>0,\,y>0,\,x^{2}+y^{2}>1\\-\pi -\arcsin \left(x\,{\sqrt {1-y^{2}}}+y\,{\sqrt {1-x^{2}}}\right),&x<0,\,y<0,\,x^{2}+y^{2}>1\end{array}}\right.\\\arcsin x-\arcsin y&=&\left\{{\begin{array}{rl}\arcsin \left(x\,{\sqrt {1-y^{2}}}-y\,{\sqrt {1-x^{2}}}\right),&xy\geq 0\;\;{\mbox{nebo}}\;\;x^{2}+y^{2}\leq 1\\\pi -\arcsin \left(x\,{\sqrt {1-y^{2}}}-y\,{\sqrt {1-x^{2}}}\right),&x>0,\,y<0,\,x^{2}+y^{2}>1\\-\pi -\arcsin \left(x\,{\sqrt {1-y^{2}}}-y\,{\sqrt {1-x^{2}}}\right),&x<0,\,y>0,\,x^{2}+y^{2}>1\end{array}}\right.\\\end{array}}}$

${\displaystyle {\begin{array}{lcll}\arcsin(\sin x)&=&x,&-{\frac {\pi }{2}}\leq x\leq {\frac {\pi }{2}}\\\sin(\arcsin x)&=&x\end{array}}}$

${\displaystyle \arcsin x=\int _{0}^{x}{\frac {\mathrm {d} t}{\sqrt {1-t^{2}}}},\quad 0<x<1}$

${\displaystyle \arcsin x={\frac {x\,{\sqrt {1-x^{2}}}}{1-\displaystyle {\frac {1\cdot 2\cdot x^{2}}{3-\displaystyle {\frac {1\cdot 2\cdot x^{2}}{5-\displaystyle {\frac {3\cdot 4\cdot x^{2}}{7-\displaystyle {\frac {3\cdot 4\cdot x^{2}}{9-\displaystyle {\frac {5\cdot 6\cdot x^{2}}{11-\dots }}}}}}}}}}}},\quad |x|<1}$

Mějme goniometrickou rovnici: [3]

${\displaystyle {\begin{array}{rcl}2\sin x&=&{\sqrt {3}}\\\hline \sin x&=&{\frac {\sqrt {3}}{2}}\\x&=&\arcsin {\frac {\sqrt {3}}{2}}\\\hline x&=&{\frac {\pi }{3}}\end{array}}}$

S ohledem na periodicitu funkce ${\displaystyle \sin x}$ jsou řešením původní rovnice také hodnoty:

${\displaystyle {\begin{array}{rrrrrrrlr}\dots ,&{\color {OliveGreen}{\frac {-11\pi }{3}}},&{\color {OliveGreen}{\frac {-5\pi }{3}}},&\mathbf {\frac {\boldsymbol {\pi }}{3}} ,&{\color {OliveGreen}{\frac {7\pi }{3}}},&{\color {OliveGreen}{\frac {13\pi }{3}}},&\dots &\qquad {\mbox{tj.}}\quad \color {OliveGreen}{x_{k}=}&\color {OliveGreen}{{\frac {\pi }{3}}+2k\pi },\;k\in \mathbb {Z} \\\dots ,&{\color {BrickRed}{\frac {-10\pi }{3}}},&{\color {BrickRed}{\frac {-4\pi }{3}}},&{\color {BrickRed}{\frac {2\pi }{3}}},&{\color {BrickRed}{\frac {8\pi }{3}}},&{\color {BrickRed}{\frac {14\pi }{3}}},&\dots &\qquad {\mbox{tj.}}\quad \color {BrickRed}{x_{k}=}&\color {BrickRed}{{\frac {2\pi }{3}}+2k\pi },\;k\in \mathbb {Z} \end{array}}}$

• Vznikne překlopením grafu funkce ${\displaystyle y=\sin(x)}$ podle osy I. a III. kvadrantu.
• Bereme pouze interval kolem počátku, na kterém je funkce ${\displaystyle \sin(x)}$ prostá, tedy v tomto případě rostoucí.
• Interval ${\displaystyle \textstyle \left\langle -{\frac {\pi }{2}},{\frac {\pi }{2}}\right\rangle }$ z definičního oboru sinu se stane oborem hodnot funkce ${\displaystyle x=\mathrm {arcsin} (y)}$.
• Obdobně obor hodnot sinu se naopak stane definičním oborem arkus sinu.

Graf funkce arkus sinus.