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Eksāmens FIZIKĀ 12. klasei
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### 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:
$±\mathrm{arccos}\phantom{\rule{0.147em}{0ex}}a+2\mathrm{\pi }n,\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\mathrm{kur}\phantom{\rule{0.147em}{0ex}}n\in \mathrm{ℤ}$

Svarīgi!
Ieteicams iegaumēt, atrodot vērtības vienības riņķī, šādiem vienādojumiem (visur atrisinājumos $$n\in\mathbb{Z}$$):
• $$\cos x=1$$. Atrisinājums ir $$x=2\pi n$$ jeb $x={360}^{o}n$

• $$\cos x=0$$. Atrisinājums ir $x=\frac{\mathrm{\pi }}{2}+\mathrm{\pi }n$ jeb $x={{90}^{o}+180}^{o}n$

• $$\cos x=-1$$. Atrisinājums ir $x=\mathrm{\pi }+2\mathrm{\pi }n$ jeb $x={180}^{o}+{360}^{o}n$

Atceries: $\mathrm{arccos}\left(-a\right)={180}^{o}-\mathrm{arccos}\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}$

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

Leņķu lielumu var izteikt ar grādiem vai ar radiāniem.

Piemērs:
Dots vienādojums $$\cos x = \frac{1}{2}$$.

Atrisinājums ir $x=\left[\begin{array}{l}\frac{\mathrm{\pi }}{3}+2\mathrm{\pi }n\\ -\frac{\mathrm{\pi }}{3}+2\mathrm{\pi }n\end{array}\right\phantom{\rule{0.147em}{0ex}},\phantom{\rule{0.294em}{0ex}}n\in \mathrm{ℤ}$

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