Razlika između verzija stranice "Kosinus"
| [pregledana izmjena] | [pregledana izmjena] |
Uklonjeni sadržaj Dodani sadržaj
m robot dodaje {{Commonscat|Cosine function}} |
No edit summary |
||
Red 9:
<math>\left[ \left( k+\frac{1}{2} \right) \pi,\; 0 \right]</math> ekstremi su tačke : <math>[k\pi,\;(-1)^n].</math>
==Osnovne osobine==
:Kosinus je neparna funkcija
:<math>\cos(-\alpha)=- cos\alpha </math>
:Kosinus je periodična funkcija
:<math>\sin(2 k\pi \pm\alpha)=sin\alpha </math>
:Nula funkcije
:<math>cos\alpha =0 = >\alpha = \frac{\pi}{2} +k \pi </math>
:Maksimum funkcije
:<math>cos\alpha =1 = >\alpha =2k\pi </math>
:Minimum funkcije
:<math>cos\alpha = -1 = >\alpha = (2k+1)\pi </math>
==Neki identiteti==
:<math> cos(\alpha + \frac{\pi}{2}) = -\sin \alpha </math>
:<math> \cos(\alpha + \pi) = -\cos \alpha </math>
:<math> cos(\alpha + 2\pi) = \cos \alpha </math>
:<math>\cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta\,</math>
:<math>\begin{align}
\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{align}</math>
:<math>\begin{align}\cos 3\alpha = \cos^3\alpha - 3 \sin^2 \alpha\cos \alpha = 4 \cos^3\alpha - 3 \cos\alpha\end{align}</math>
:<math>\cos \frac{\alpha}{2} = \pm\, \sqrt{\frac{1 + \cos\alpha}{2}}</math>
:<math>\cos^2\alpha = \frac{1 + \cos 2\alpha}{2}\!</math>
:<math>\cos \alpha \cos \beta = {\cos(\alpha - \beta + \cos(\alpha + \beta) \over 2}</math>
:<math>\cos \alpha + \cos \beta = 2 \cos\left( \frac{\alpha + \beta} {2} \right) \cos\left( \frac{\alpha - \beta}{2} \right)</math>
:<math>\cos \alpha - \cos \beta = -2 \sin\left( \frac{\alpha + \beta} {2} \right) \sin\left( \frac{\alpha - \beta}{2} \right)</math>
: <math>\cos(\alpha) = \frac{e^{ix} + e^{-ix}}{2} \;</math> za <math>i^2=-1</math>
:<math>\cos x = \prod_{n = 1}^\infty\left(1 - \frac{x^2}{\pi^2(n - \frac{1}{2})^2}\right)</math>
Zlatni rez
;<math>\cos \left( \frac {\pi} {5} \right) = \cos 36^\circ={\sqrt{5}+1 \over 4} = \frac{\varphi }{2}
</math>
==Derivacije==
:<math>\lim_{x\rightarrow 0}\frac{1-\cos x }{x}=0,</math>
:<math>(cos x)' =- \sin x </math>
==Inverzna funkcija==
Inverzna funkcija funkcije
:<math>\cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2} \,</math> je funcija
:<math>\arccos x = -i \ln \left(x + \sqrt{x^2 - 1}\right) \,</math>
Koristi se za određivanje veličine ugla , kada je poznata vrijednost njegovog kosinusa.
:<math>\ arccos(x) = \frac{\pi}{2} - \arcsin(x)</math>
:<math>\arccos(-x) = \pi -\arccos(x) </math>
:<math>
\arccos\left(\frac{1}{x}\right) = \arcsec(x) </math>
:<math>
\arccos(x) = \arcsin\left(\sqrt{1 - x^2}\right) \, , \text{ if } 0 \leq x \leq 1 </math>
:<math>\arccos(x) = \frac{1}{2}\arccos\left(2x^2-1\right) \, , \text{ if } 0 \leq x \leq 1
</math>
:<math>\arccos(x) = 2 \arctan\left(\frac{\sqrt{1 - x^2}}{1 + x}\right) \, , \text{ if } -1 < x \leq + 1 </math>
: <math>\frac{d}{dx} \arcsin(z) = \frac{1}{\sqrt{1-z^2}} </math>
: <math>\arccos(x) = \int_x^1 \frac{1}{\sqrt{1 - z^2}} \, |x| {} \leq 1 </math>
:<math>\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| \le 1</math>
== Također pogledajte ==
Line 17 ⟶ 75:
* [[Kotangens]]
{{Commonscat|Cosine function}}
[[Kategorija:Trigonometrija]]
| |||