### Teorija

Vienādojuma cosx = a atrisināšana

Vienādojumam cosx = a  eksistē atrisinājums, ja $-1\le a\le 1\phantom{\rule{0.588em}{0ex}}\mathit{jeb}\phantom{\rule{0.147em}{0ex}}\left|a\right|\le 1$

x = $\left[\begin{array}{l}\mathit{arccosa}+{360}^{o}n\\ -\mathit{arccosa}+{360}^{o}n\end{array}\right\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\mathit{jeb}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\left[\begin{array}{l}\mathit{arccosa}+\mathit{2}\mathrm{\pi }n\\ -\mathit{arccosa}+\mathit{2}\mathrm{\pi }n\end{array}\right\phantom{\rule{0.147em}{0ex}},\phantom{\rule{0.147em}{0ex}}\mathit{kur}\phantom{\rule{0.147em}{0ex}}n\in Z$

Atbildes var apvienot vienā :  $±\mathit{arccosa}+2\mathrm{\pi }n,\phantom{\rule{0.294em}{0ex}}\mathit{kur}\phantom{\rule{0.294em}{0ex}}n\in Z$

Svarīgi!
Ieteicams atrast šīs vērtības vienības riņķī un iegaumēt:
$\begin{array}{l}\mathit{cosx}=1\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}x={360}^{o}n\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\mathit{jeb}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}x=2\mathrm{\pi }n\\ \mathit{cosx}=0\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}x={{90}^{o}+180}^{o}n\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\mathit{jeb}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}x=\frac{\mathrm{\pi }}{2}+\mathrm{\pi }n\\ \\ \mathit{cosx}=-1\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}x={180}^{o}+{360}^{o}n\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\mathit{jeb}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}x=\mathrm{\pi }+2\mathrm{\pi }n\\ \phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\end{array}$

Atceries:  $\mathit{arccos}\left(-a\right)={180}^{o}-\mathit{arccosa}\phantom{\rule{0.441em}{0ex}}$

x = $\left[\begin{array}{l}180\mathrm{°}-\mathit{arccosa}+{360}^{o}n\\ 180\mathrm{°}+\mathit{arccosa}+{360}^{o}n\end{array}\right\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\mathit{jeb}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\left[\begin{array}{l}\mathrm{\pi }-\mathit{arccosa}+\mathit{2}\mathrm{\pi }n\\ \mathrm{\pi }+\mathit{arccosa}+\mathit{2}\mathrm{\pi }n\end{array}\right\phantom{\rule{0.147em}{0ex}},\phantom{\rule{0.147em}{0ex}}\mathit{kur}\phantom{\rule{0.147em}{0ex}}n\in Z$

Leņķu lielumu var izteikt ar grādiem vai ar radiāniem.

Piemērs:
$\begin{array}{l}\mathit{cosx}=\frac{1}{2}\\ x=\left[\begin{array}{l}{60}^{o}+{360}^{o}n\\ -{60}^{o}+{360}^{o}n\end{array}\right\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\mathit{jeb}\phantom{\rule{0.147em}{0ex}}x=\left[\begin{array}{l}\frac{\mathrm{\pi }}{3}+2\mathrm{\pi }n\\ -\frac{\mathrm{\pi }}{3}+2\mathrm{\pi }n,\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\mathit{kur}\phantom{\rule{0.147em}{0ex}}n\in Z\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\end{array}\right\end{array}$

Piemērs:
$\begin{array}{l}\mathit{cosx}=-\frac{1}{2}\\ x=\left[\begin{array}{l}{{180}^{o}-60}^{o}+{360}^{o}n\\ {180}^{o}+{60}^{o}+{360}^{o}n\end{array}\right\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}x=\left[\begin{array}{l}{120}^{o}+{360}^{o}n\\ {240}^{o}+{360}^{o}n\end{array}\right\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}n\in Z\end{array}$

Piemērs:
$\begin{array}{l}\mathit{cosx}=-0,3\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ x=\left[\begin{array}{l}\mathit{arccos}\left(-0,3\right)+{360}^{o}n\\ -\mathit{arccos}\left(-0,3\right)+{360}^{o}n\end{array}\right\phantom{\rule{1.176em}{0ex}}x=\left[\begin{array}{l}{180}^{o}-\mathit{arccos0},3+{360}^{o}n\\ {180}^{o}+\mathit{arccos0},3+{360}^{o}n\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}n\in Z\end{array}\right\end{array}$

Piemērs:
$\begin{array}{l}\mathit{cosx}=-4\\ \\ \phantom{\rule{0.147em}{0ex}}\mathit{jo}\phantom{\rule{0.147em}{0ex}}-4<-1\end{array}$ sakņu nav,  cos vērtību apgabals ir [-1;1]