"MATEMĀTIKA 11. KLASEI"
Aplūkosim trigonometriskā vienādojuma piemēru, kurā ir dota triju dažādu funkciju summa.
Visvienkāršāk ir pārveidot tangensa funkciju. Izmantojam formulu.
Piemērs:
Atrisini vienādojumu $\mathrm{sin}x-4\mathrm{cos}x+\mathrm{tg}x=4$

Risinājums
Pēc formulas pārveido $\mathrm{tg}x=\frac{\mathrm{sin}x}{\mathrm{cos}x}$.
Nosaka kopsaucēju:
$\begin{array}{l}{\frac{\mathrm{sin}x}{1}}^{\left(\mathrm{cos}x}\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}{\frac{4\mathrm{cos}x}{1}}^{\left(\mathrm{cos}x}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\frac{\mathrm{sin}x}{\mathrm{cos}x}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}{\frac{4}{1}}^{\left(\mathrm{cos}x}\\ \frac{\mathrm{sin}x\mathrm{cos}x-4{\mathrm{cos}}^{2}x+\mathrm{sin}x}{\mathrm{cos}x}=\frac{4\mathrm{cos}x}{\mathrm{cos}x}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ \mathrm{cos}x\ne 0\end{array}$

Ja saucēji vienādi, arī skaitītāji ir vienādi. Izmanto grupēšanu.

$\begin{array}{l}\mathrm{sin}x\mathrm{cos}x-4{\mathrm{cos}}^{2}x+\mathrm{sin}x-4\mathrm{cos}x=0\\ \mathrm{sin}x\mathrm{cos}x+\mathrm{sin}x-4{\mathrm{cos}}^{2}x-4\mathrm{cos}x=0\\ \\ \mathrm{sin}x\left(\mathrm{cos}x+1\right)-4\mathrm{cos}x\left(\mathrm{cos}x+1\right)=0\\ \left(\mathrm{cos}x+1\right)\left(\mathrm{sin}x-4\mathrm{cos}x\right)=0\end{array}$

Reizinājums ir vienāds ar $$0$$, ja vismaz viens no reizinātājiem ir vienāds ar $$0$$.
$\begin{array}{l}\mathrm{cos}x+1=0\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ \mathrm{cos}x=-1\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ x=\mathrm{\pi }+2\mathrm{\pi }k,\phantom{\rule{0.147em}{0ex}}k\in \mathrm{ℤ}\\ \mathrm{vai}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ \mathrm{sin}x-4\mathrm{cos}x=0\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}|:\mathrm{cos}x\ne 0\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ \frac{\mathrm{sin}x}{\mathrm{cos}x}-4=0\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ \mathrm{tg}x-4=0\\ \mathrm{tg}x=4\\ x=\mathrm{arctg}4+\mathrm{\pi }n\phantom{\rule{0.147em}{0ex}},\phantom{\rule{0.147em}{0ex}}n\in \mathrm{ℤ}\end{array}$

Pārbauda, vai iegūtās saknes ir derīgas ($\mathrm{cos}x$ nedrīkst būt $$0$$, jo ir daļas saucējā)

Atbilde:
$\begin{array}{l}x=\mathrm{\pi }+2\mathrm{\pi }k,\phantom{\rule{0.147em}{0ex}}k\in \mathrm{ℤ}\\ x=\mathrm{arctg}4+\mathrm{\pi }n,\phantom{\rule{0.147em}{0ex}}n\in \mathrm{ℤ}\end{array}$
Izmēģini iegūtās zināšanas! Uz lapas izrēķini vienādojumu $\mathrm{cos}x-8\mathrm{sin}x+\mathrm{ctg}x=8$.
Salīdzini, vai ieguvi atbildi:
$\begin{array}{l}x=-\frac{\mathrm{\pi }}{2}+2\mathrm{\pi }k,\phantom{\rule{0.147em}{0ex}}k\in \mathrm{ℤ}\\ x=\mathrm{arcctg}8+\mathrm{\pi }n,\phantom{\rule{0.147em}{0ex}}n\in \mathrm{ℤ}\end{array}$