### Teorija

Vienādojumam $$\cos x = a$$ eksistē atrisinājums, ja $$-1\leq a\leq 1$$ jeb $$|a|\leq 1$$.

Ja $$\cos x = a$$, tad
$x=\left[\begin{array}{l}\mathrm{arccos}\phantom{\rule{0.147em}{0ex}}a+2\mathrm{\pi }n\\ -\mathrm{arccos}\phantom{\rule{0.147em}{0ex}}a+2\mathrm{\pi }n\end{array}\right\phantom{\rule{0.147em}{0ex}},\phantom{\rule{0.147em}{0ex}}\mathrm{kur}\phantom{\rule{0.147em}{0ex}}n\in \mathrm{ℤ}$

($$2\pi$$ atbilst $$360$$ grādiem.)

Šīs atbildes var apvienot vienā:
$x=±\mathrm{arccos}\phantom{\rule{0.147em}{0ex}}a+2\mathrm{\pi }n,\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\mathrm{kur}\phantom{\rule{0.147em}{0ex}}n\in \mathrm{ℤ}$
Arkkosinuss no skaitļa $$a$$, ir tas pagrieziena leņķis no intervāla $\left[0;\mathrm{\pi }\right]$, kura kosinuss ir vienāds ar skaitli $$a$$.
$\mathit{arccos}\frac{1}{2}=\frac{\mathrm{\pi }}{3},\phantom{\rule{0.294em}{0ex}}\mathit{jo}\phantom{\rule{0.147em}{0ex}}\mathit{cos}\frac{\mathrm{\pi }}{3}=\frac{1}{2}\phantom{\rule{0.147em}{0ex}}\mathit{un}\phantom{\rule{0.147em}{0ex}}\frac{\mathrm{\pi }}{3}\in \left[0;\mathrm{\pi }\right]$
$\mathrm{arccos}\left(-a\right)=\mathrm{\pi }-\mathrm{arccos}\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}$
$\mathit{arccos}\left(-\frac{1}{2}\right)=\mathrm{\pi }-\mathit{arccos}\frac{1}{2}=\mathrm{\pi }-\frac{\mathrm{\pi }}{3}=\frac{2\mathrm{\pi }}{3}$, ja leņķi pieraksta ar radiāniem.

$\mathit{arccos}\left(-\frac{1}{2}\right)=180\mathrm{°}-\mathit{arccos}\frac{1}{2}=180\mathrm{°}-60\mathrm{°}=120\mathrm{°}$, ja leņķi pieraksta ar grādiem.

Ja kosinusa vērtība ir negatīva: $$\cos x=-a$$, tad
$x=\phantom{\rule{0.147em}{0ex}}\left[\begin{array}{l}\mathrm{\pi }-\mathrm{arccos}\phantom{\rule{0.147em}{0ex}}a+2\mathrm{\pi }n\\ \mathrm{\pi }+\mathrm{arccos}\phantom{\rule{0.147em}{0ex}}a+2\mathrm{\pi }n\end{array}\right\phantom{\rule{0.147em}{0ex}},\phantom{\rule{0.147em}{0ex}}\mathrm{kur}\phantom{\rule{0.147em}{0ex}}n\in \mathrm{ℤ}$
Piemērs:
Dots vienādojums $$\cos x = \frac{1}{2}$$.
Atrisinājums ir $x=\left[\begin{array}{l}\frac{\mathrm{\pi }}{3}+2\mathrm{\pi }n\\ -\frac{\mathrm{\pi }}{3}+2\mathrm{\pi }n\end{array}\right\phantom{\rule{0.147em}{0ex}},\phantom{\rule{0.294em}{0ex}}n\in \mathrm{ℤ}$

$x=\left[\begin{array}{l}{60}^{o}+{360}^{o}n\\ -{60}^{o}+{360}^{o}n\end{array}\right\phantom{\rule{0.147em}{0ex}},\phantom{\rule{0.294em}{0ex}}n\in \mathrm{ℤ}$
$\begin{array}{l}\mathrm{cos}\phantom{\rule{0.147em}{0ex}}x=-\frac{1}{2}\\ x=\left[\begin{array}{l}{{180}^{o}-60}^{o}+{360}^{o}n\\ {180}^{o}+{60}^{o}+{360}^{o}n\end{array}\right\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}},\phantom{\rule{0.147em}{0ex}}n\in \mathrm{ℤ}\\ x=\left[\begin{array}{l}{120}^{o}+{360}^{o}n\\ {240}^{o}+{360}^{o}n\end{array}\right\phantom{\rule{0.147em}{0ex}},\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}n\in \mathrm{ℤ}\end{array}$
$\begin{array}{l}\mathrm{cos}\phantom{\rule{0.147em}{0ex}}x=-0,3\\ \phantom{\rule{0.147em}{0ex}}\\ x=\left[\begin{array}{l}\mathrm{arccos}\left(-0,3\right)+{360}^{o}n\\ -\mathrm{arccos}\left(-0,3\right)+{360}^{o}n\end{array}\right\phantom{\rule{0.147em}{0ex}},\phantom{\rule{0.294em}{0ex}}n\in \mathrm{ℤ}\\ x=\left[\begin{array}{l}{180}^{o}-\mathrm{arccos}\phantom{\rule{0.147em}{0ex}}0,3+{360}^{o}n\\ {180}^{o}+\mathrm{arccos}\phantom{\rule{0.147em}{0ex}}0,3+{360}^{o}n\end{array}\right\phantom{\rule{0.147em}{0ex}},\phantom{\rule{0.294em}{0ex}}n\in \mathrm{ℤ}\end{array}$
Vienādojumam $$\cos x=-4$$ sakņu nav, jo kosinusa vērtību apgabals ir $$[-1;1]$$, bet $$-4<-1$$.