Kosinus

Kosinus je parna trigonometrijska funkcija oblika:

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

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

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

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

Osnovne osobine

Kosinus je parna funkcija

$\cos(-\alpha )=-cos\alpha$

Kosinus je periodična funkcija

$\sin(2k\pi \pm \alpha )=sin\alpha$

Nula funkcije

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

Maksimum funkcije

$cos\alpha =1=>\alpha =2k\pi$

Minimum funkcije

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

Neki identiteti

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

$\cos(\alpha +\pi )=-\cos \alpha$

$cos(\alpha +2\pi )=\cos \alpha$

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

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

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

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

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

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

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

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

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

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

Zlatni rez

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

Derivacije

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

$(cosx)'=-\sin x$

Inverzna funkcija

Inverzna funkcija funkcije

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

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

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

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

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

$\arccos(-x)=\pi -\arccos(x)$

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

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

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

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

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

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

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

Također pogledajte

• Kosinusoida
• Sinus
• Tangens
• Kotangens