Kosinus

Kosinus je parna trigonometrijska funkcija oblika:

${\displaystyle y=a\cos(\omega x+\phi );\,}$. Ova jednačina ima i oblik:

${\displaystyle y=a\sin(\omega x+\phi +{\frac {\pi }{2}}).}$ .

Grafik ove funkcije naziva se kosinusoida, koji presjeca x-osu u

${\displaystyle \left[\left(k+{\frac {1}{2}}\right)\pi ,\;0\right]}$ ekstremi su tačke : ${\displaystyle [k\pi ,\;(-1)^{n}].}$

Osnovne osobine

Kosinus je parna funkcija

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

Kosinus je periodična funkcija

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

Nula funkcije

${\displaystyle cos\alpha =0=>\alpha ={\frac {\pi }{2}}+k\pi }$

Maksimum funkcije

${\displaystyle cos\alpha =1=>\alpha =2k\pi }$

Minimum funkcije

${\displaystyle cos\alpha =-1=>\alpha =(2k+1)\pi }$

Neki identiteti

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

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

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

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

${\displaystyle {\begin{aligned}\cos 2\alpha =\cos ^{2}\alpha -\sin ^{2}\alpha =2\cos ^{2}\alpha -1=1-2\sin ^{2}\alpha ={\frac {1-\tan ^{2}\alpha }{1+\tan ^{2}\alpha }}\end{aligned}}}$

${\displaystyle {\begin{aligned}\cos 3\alpha =\cos ^{3}\alpha -3\sin ^{2}\alpha \cos \alpha =4\cos ^{3}\alpha -3\cos \alpha \end{aligned}}}$

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

${\displaystyle \cos ^{2}\alpha ={\frac {1+\cos 2\alpha }{2}}\!}$

${\displaystyle \cos \alpha \cos \beta ={\cos(\alpha -\beta +\cos(\alpha +\beta ) \over 2}}$

${\displaystyle \cos \alpha +\cos \beta =2\cos \left({\frac {\alpha +\beta }{2}}\right)\cos \left({\frac {\alpha -\beta }{2}}\right)}$

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

${\displaystyle \cos(\alpha )={\frac {e^{ix}+e^{-ix}}{2}}\;}$ za ${\displaystyle i^{2}=-1}$

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

Zlatni rez

${\displaystyle \cos \left({\frac {\pi }{5}}\right)=\cos 36^{\circ }={{\sqrt {5}}+1 \over 4}={\frac {\varphi }{2}}}$

Derivacije

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

${\displaystyle (cosx)'=-\sin x}$

Inverzna funkcija

Inverzna funkcija funkcije

${\displaystyle \cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}\,}$ je funcija

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

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

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

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

${\displaystyle \arccos(-x)=\pi -\arccos(x)}$

${\displaystyle \arccos \left({\frac {1}{x}}\right)=\operatorname {arcsec}(x)}$

${\displaystyle \arccos(x)=\arcsin \left({\sqrt {1-x^{2}}}\right)\,,{\text{ if }}0\leq x\leq 1}$

${\displaystyle \arccos(x)={\frac {1}{2}}\arccos \left(2x^{2}-1\right)\,,{\text{ if }}0\leq x\leq 1}$

${\displaystyle \arccos(x)=2\arctan \left({\frac {\sqrt {1-x^{2}}}{1+x}}\right)\,,{\text{ if }}-1<x\leq +1}$

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

${\displaystyle \arccos(x)=\int _{x}^{1}{\frac {1}{\sqrt {1-z^{2}}}}\,|x|{}\leq 1}$

${\displaystyle \arccos(z)={\frac {\pi }{2}}-\arcsin(z)={\frac {\pi }{2}}-\left(z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\cdots \right)={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {{\binom {2n}{n}}z^{2n+1}}{4^{n}(2n+1)}}\,;\qquad |z|\leq 1}$

Također pogledajte

• Kosinusoida
• Sinus
• Tangens
• Kotangens