### Teorija

Lietojot argumentu saskaitīšanas formulas, var iegūt dubultleņķu formulas, pēc kurām funkcijas $$\sin 2x$$, $$\cos 2x$$, $$\operatorname{tg}2x$$ var izteikt ar leņķa $$x$$ funkcijām.

Zināms, ka $$\sin(x+y)=\sin x\cdot \cos y + \cos x \cdot \sin y$$.
Ja $$x=y$$, tad $$\sin(x+x)=\sin x\cdot \cos x+\cos x\cdot \sin x$$.
$$\sin 2x=\sin x\cdot \cos x+\cos x\cdot \sin x$$
Divkārša leņķa sinuss ir vienāds ar divkāršotu leņķa sinusa un leņķa kosinusa reizinājumu.
Identitātē $$\cos(x+y)=\cos x\cdot \cos y - \sin x \cdot \sin y$$, ievietojot $$x=y$$, iegūst
$$\cos(x+x)=\cos x\cdot \cos x - \sin x\cdot \sin x$$
$$\cos 2x=\cos^2 x - \sin^2 x$$
Divkārša leņķa kosinuss ir vienāds ar starpību starp leņķa kosinusa kvadrātu un leņķa sinusa kvadrātu.
Līdzīgi iegūst arī formulu $$\operatorname{tg} 2x$$ izteikšanai.

$\mathrm{tg}\phantom{\rule{0.147em}{0ex}}\left(x+y\right)\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{\mathrm{tg}\phantom{\rule{0.147em}{0ex}}x\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\mathrm{tg}\phantom{\rule{0.147em}{0ex}}y}{1\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}\mathrm{tg}\phantom{\rule{0.147em}{0ex}}x\phantom{\rule{0.147em}{0ex}}\cdot \phantom{\rule{0.147em}{0ex}}\mathrm{tg}\phantom{\rule{0.147em}{0ex}}y}$

Ja $$x=y$$, tad

$\begin{array}{l}\mathrm{tg}\phantom{\rule{0.147em}{0ex}}\left(x+x\right)\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{\mathrm{tg}\phantom{\rule{0.147em}{0ex}}x\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\mathrm{tg}\phantom{\rule{0.147em}{0ex}}x}{1\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}\mathrm{tg}\phantom{\rule{0.147em}{0ex}}x\phantom{\rule{0.147em}{0ex}}\cdot \phantom{\rule{0.147em}{0ex}}\mathrm{tg}\phantom{\rule{0.147em}{0ex}}x}\\ \\ \mathrm{tg}\phantom{\rule{0.147em}{0ex}}2x\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{2\phantom{\rule{0.147em}{0ex}}\mathrm{tg}\phantom{\rule{0.147em}{0ex}}x}{1\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}{\mathrm{tg}}^{2}x}\end{array}$
Piemērs:
Divkāršā leņķa formulu ir izdevīgi pielietot, lai divu funkciju reizinājumu izteiktu kā vienu funkciju.
$$\sin x\cdot \cos x=\frac{1}{2}\cdot 2 \sin x\cdot \cos x = \frac{1}{2} \sin 2x$$