Aplūkosim trigonometriskā vienādojuma piemēru, kurā ir dota triju dažādu funkciju summa.
Visvienkāršāk ir pārveidot tangensa funkciju. Izmantojam formulu.
Piemērs:
Atrisini vienādojumu $$sinx - 4cosx + tgx = 4$$

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

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

$\begin{array}{l}\mathit{sinxcosx}-4{\mathit{cos}}^{2}x+\mathit{sinx}-4\mathit{cosx}=0\\ \mathit{sinxcosx}+\mathit{sinx}-4{\mathit{cos}}^{2}x-4\mathit{cosx}=0\\ \\ \mathit{sinx}\left(\mathit{cosx}+1\right)-4\mathit{cosx}\left(\mathit{cosx}+1\right)=0\\ \left(\mathit{cosx}+1\right)\left(\mathit{sinx}-4\mathit{cosx}\right)=0\end{array}$

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

Pārbauda, vai iegūtās saknes ir derīgas ($$cosx$$ nedrīkst būt $$0$$, jo ir daļas saucējā)

Atbilde:
$\begin{array}{l}x=\mathrm{\pi }+2\mathrm{\pi }k,\phantom{\rule{0.147em}{0ex}}k\in \mathrm{ℤ}\\ x=\mathit{arctg}4+\mathrm{\pi }n,\phantom{\rule{0.147em}{0ex}}n\in \mathrm{ℤ}\end{array}$
Izmēģini iegūtās zināšanas! Uz lapas izrēķini vienādojumu $\mathit{cosx}-8\mathit{sinx}+\mathit{ctgx}=8$.
Salīdzini, vai ieguvi atbildi:
$\begin{array}{l}x=-\frac{\mathrm{\pi }}{2}+2\mathrm{\pi }k,\phantom{\rule{0.147em}{0ex}}k\in \mathrm{ℤ}\\ x=\mathit{arcctg}8+\mathrm{\pi }n,\phantom{\rule{0.147em}{0ex}}n\in \mathrm{ℤ}\end{array}$
Atsauce:
Materiālu sagatavoja. Mg. math. Laima Baltiņa