Sinus

Sinus je neparna funkcija

${\displaystyle \sin(-\alpha )=-sin\alpha }$

Sinus je periodična funkcija

${\displaystyle \sin(2k\pi \pm \alpha )=sin\alpha }$

Nula funkcije

${\displaystyle sin\alpha =0=>\alpha =k\pi }$

Maksimum funkcije

${\displaystyle sin\alpha =1=>\alpha =2k\pi +{\frac {\pi }{2}}}$

Minimum funkcije

${\displaystyle sin\alpha =-1=>\alpha =2k\pi -{\frac {\pi }{2}}}$

${\displaystyle sin(\alpha +{\frac {\pi }{2}})=\cos \alpha }$

${\displaystyle sin(\alpha +\pi )=-\sin \alpha }$

${\displaystyle :sin(\alpha +2\pi )=\sin \alpha }$

${\displaystyle \sin(\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta \!}$

${\displaystyle sin2\alpha =2\sin \alpha \cos \alpha ={\frac {2\tan \alpha }{1+\tan ^{2}\alpha }}}$

${\displaystyle sin3\alpha =3\sin \alpha -4\sin ^{3}\alpha }$

${\displaystyle sin{\frac {\alpha }{2}}=\pm \,{\sqrt {\frac {1-\cos \alpha }{2}}}}$

${\displaystyle \sin \alpha \pm \sin \beta =2\sin \left({\frac {\alpha \pm \beta }{2}}\right)\cos \left({\frac {\alpha \mp \beta }{2}}\right)}$

${\displaystyle {\text{Ako su }}w+x+y+z=\pi ={\text{polukrug,}}\,}$

${\displaystyle {\begin{aligned}{\text{tada vrijedi }}&\sin(w+x)\sin(x+y)\\&{}=\sin(x+y)\sin(y+z)\\&{}=\sin(y+z)\sin(z+w)\\&{}=\sin(z+w)\sin(w+x)=\sin(w)\sin(y)+\sin(x)\sin(z).\end{aligned}}}$

${\displaystyle \sin(x)={\frac {e^{ix}-e^{-ix}}{2i}}}$ ( ${\displaystyle i^{2}=-1}$ )

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

${\displaystyle |\sin x|={\frac {1}{2}}\prod _{n=0}^{\infty }{\sqrt[{2^{n+1}}]{\left|\tan \left(2^{n}x\right)\right|}}}$

Zlatni rez

${\displaystyle \sin \left({\frac {\pi }{10}}\right)=\sin 18^{\circ }={{\sqrt {5}}-1 \over 4}={\varphi -1 \over 2}={1 \over 2\varphi }}$

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

Izvod

${\displaystyle (sinx)'=\cos x}$

Integral

${\displaystyle \int {\frac {du}{\sqrt {a^{2}-u^{2}}}}=\sin ^{-1}\left({\frac {u}{a}}\right)+C}$

Inverzna funkcija funkcije ${\displaystyle \sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}\,}$ je

${\displaystyle \mathrm {e} ^{\mathrm {i} \theta }-\mathrm {e} ^{-\mathrm {i} \theta }=(\cos \theta +\mathrm {i} \sin \theta )-(\cos \theta -\mathrm {i} \sin \theta )=2\mathrm {i} \cdot \sin \theta \Rightarrow \sin \theta ={\mathrm {e} ^{\mathrm {i} \theta }-\mathrm {e} ^{-\mathrm {i} \theta } \over 2\mathrm {i} }}$

${\displaystyle \arcsin x=-i\ln \left(ix+{\sqrt {1-x^{2}}}\right)\,}$

${\displaystyle \arcsin \alpha }$ (Arkus sinus) je funkcija inverzna funkciji ${\displaystyle \sin \alpha }$ na njenom intervalu ${\displaystyle [-{\frac {\pi }{2}},{\frac {\pi }{2}}]}$

Koristi se za određivanje veličine ugla , kada je poznata vrijednost njegovog sinusa.

Formule

${\displaystyle \arcsin {-x}={\frac {\pi }{2}}-\arccos {x}}$ (pravilo komplementarnih uglova)

${\displaystyle \arcsin {-x}=-\arcsin {x}}$ (neparnost funkcije)

${\displaystyle \arcsin {\frac {1}{x}}=arccosec{x}}$

${\displaystyle \arcsin x=2arctg{\frac {x}{1+{\sqrt {1-x^{2}}}}}}$

${\displaystyle (arcsin)'={\frac {1}{\sqrt {1-x^{2}}}};\qquad |x|<1}$

${\displaystyle \arcsin x{}=\int _{0}^{x}{\frac {1}{\sqrt {1-z^{2}}}}\,dz,\qquad |x|\leq 1}$

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

• Sinusoida
• Kosinus
• Tangens
• Kotangens

1. Sinus, mathworld.wolfram.com
2. Trigonometrijske funkcije realnog broja Arhivirano 28. 3. 2016. na Wayback Machine