### Teorija

Vienādojuma sinx = a atrisināšana

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

Leņķus var izteikt grādos vai radiānos.

x = $\left[\begin{array}{l}\mathit{arcsina}+{360}^{o}n\\ {180}^{o}-\mathit{arcsina}+{360}^{o}n\end{array}\right\phantom{\rule{0.882em}{0ex}}\mathit{jeb}\phantom{\rule{0.294em}{0ex}}\left[\begin{array}{l}\mathit{arcsina}+\mathit{2}\mathrm{\pi }n\\ \mathrm{\pi }-\mathit{arcsina}+\mathit{2}\mathrm{\pi }n\end{array}\right\phantom{\rule{0.147em}{0ex}},\phantom{\rule{0.147em}{0ex}}\mathit{kur}\phantom{\rule{0.147em}{0ex}}n\in Z$

Atbildes var apvienot vienā :  ${\left(-1\right)}^{n}\cdot \mathit{arcsina}+\mathrm{\pi }n,\phantom{\rule{0.294em}{0ex}}\mathit{kur}\phantom{\rule{0.147em}{0ex}}n\in Z$ , bet šādu atbildi ir grūtāk izprast un redzēt saistību ar trigonometriskajām vērtībām vienības riņķī.

Svarīgi!
Ieteicams atrast šīs vērtības vienības riņķī un iegaumēt:
$\begin{array}{l}\mathit{sinx}=1\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}x={90}^{o}+{360}^{o}n\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.882em}{0ex}}\phantom{\rule{0.147em}{0ex}}\mathit{jeb}\phantom{\rule{0.147em}{0ex}}x=\frac{\mathrm{\pi }}{2}+2\mathrm{\pi }n\\ \mathit{sinx}=0\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}x={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}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.882em}{0ex}}\phantom{\rule{0.147em}{0ex}}\mathit{jeb}\phantom{\rule{0.147em}{0ex}}x=\mathrm{\pi }n\\ \\ \mathit{sinx}=-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={270}^{o}+{360}^{o}n\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=\frac{3\mathrm{\pi }}{2}+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}}\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{vai}\\ \mathit{sinx}=-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=-{90}^{o}+{360}^{o}n\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=-\frac{\mathrm{\pi }}{2}+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}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\end{array}$

Atceries: arcsin(-a) = -arcsina

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

Piemērs:
$\begin{array}{l}\mathit{sinx}=\frac{\sqrt{3}}{2}\\ x=\left[\begin{array}{l}{60}^{o}+{360}^{o}n\\ {180}^{o}-{60}^{o}+{360}^{o}n\end{array}\right\phantom{\rule{1.176em}{0ex}}x=\left[\begin{array}{l}{60}^{o}+{360}^{o}n\\ {120}^{o}+{360}^{o}n\end{array}\right\phantom{\rule{0.588em}{0ex}}n\in Z\end{array}$

Piemērs:
$\begin{array}{l}\mathit{sinx}=-\frac{\sqrt{3}}{2}\\ x=\left[\begin{array}{l}{-60}^{o}+{360}^{o}n\\ {180}^{o}+{60}^{o}+{360}^{o}n\end{array}\right\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}x=\left[\begin{array}{l}{-60}^{o}+{360}^{o}n\\ {240}^{o}+{360}^{o}n\end{array}\right\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}n\in Z\end{array}$

Piemērs:
$\begin{array}{l}\mathit{sinx}=0,2\phantom{\rule{1.323em}{0ex}}\\ x=\left[\begin{array}{l}\mathit{arcsin0},2+{360}^{o}n\\ {180}^{o}-\mathit{arcsin0},2+{360}^{o}n\phantom{\rule{1.176em}{0ex}}n\in Z\end{array}\right\end{array}$

Piemērs:
$\begin{array}{l}\mathit{sinx}=\sqrt{7}\\ \\ \phantom{\rule{0.147em}{0ex}}\mathit{jo}\phantom{\rule{0.147em}{0ex}}\sqrt{7}>1\end{array}$ sakņu nav, sin funkcijas vērtību apgabals ir [-1;1]

$n\in Z$ nozīmē, ka n vērtības ir visi veselie skaitļi.