Výuka
Pokud není řečeno jinak, všechny níže uvedené vztahy a vzorce jsou platné, patří-li $x$ a $y$ do definičního oboru příslušných funkcí a mají-li dané výrazy smysl.
\begin{eqnarray} D(\sin x) & = & \R \\ D(\cos x) & = & \R \\ D(\tg x) & = & \R - \left\{\frac{\pi}{2}+k\pi,\ k\in\Z\right\} \\ D(\cotg x) & = & \R - \left\{k\pi,\ k\in\Z\right\} \\ \sin(-x) & = & -\sin x \qquad (\text{lichá funkce}) \\ \cos(-x) & = & \cos x \qquad\;\;\;\; (\text{sudá funkce}) \\ \tg(-x) & = & -\tg x \qquad\;\;\; (\text{lichá funkce}) \\ \cotg(-x) & = & -\cotg x \qquad (\text{lichá funkce}) \\ \end{eqnarray} \begin{eqnarray} \sin(x+2k\pi) & = & \sin x \qquad (\text{perioda je}\ 2\pi) \\ \cos(x+2k\pi) & = & \cos x \qquad (\text{perioda je}\ 2\pi) \\ \tg(x+k\pi) & = & \tg x \qquad\;\;\; (\text{perioda je}\ \pi) \\ \cotg(x+k\pi) & = & \cotg x \qquad (\text{perioda je}\ \pi) \\ \sin\left(x+\frac{\pi}{2}\right) & = & \cos x \\ \cos\left(x-\frac{\pi}{2}\right) & = & \sin x \\ \end{eqnarray} \begin{eqnarray} \tg x & = & \frac{\sin x}{\cos x} \;\; = \;\; \frac{1}{\cotg x} \\ \cotg x & = & \frac{\cos x}{\sin x} \;\;=\;\; \frac{1}{\tg x} \\ \end{eqnarray}
\begin{eqnarray} \sin x+\sin y & = & 2\sin\left(\frac{x + y}{2}\right)\cos\left(\frac{ x- y}{2}\right) \\ \sin x-\sin y & = & 2\sin\left(\frac{x - y}{2}\right)\cos\left(\frac{ x+ y}{2}\right) \\ \cos x+\cos y & = & 2\cos\left(\frac{x + y}{2}\right)\cos\left(\frac{ x- y}{2}\right) \\ \cos x-\cos y & = & -2\sin\left(\frac{x - y}{2}\right)\sin\left(\frac{ x+ y}{2}\right) \\ \sin x+\cos x & = & \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) \; = \; \sqrt{2} \cos\left( x-\frac{\pi}{4}\right) \\ \sin x-\cos x & = & \sqrt{2} \sin\left(x - \frac{\pi}{4}\right) \; = -\sqrt{2} \cos\left( x+\frac{\pi}{4}\right) \\ \end{eqnarray} \begin{eqnarray} \sin( x+ y) & = & \sin x\cos y+\sin y\cos x \label{sp} \\ \sin( x- y) & = & \sin x\cos y-\sin y\cos x \\ \cos( x+ y) & = & \cos x\cos y-\sin x\sin y \label{cp} \\ \cos( x- y) & = & \cos x\cos y+\sin x\sin y \label{cm}\\ \tg( x+ y) & = & \frac{\tg x+\tg y}{1-\tg x\tg y} \\ \tg( x- y) & = & \frac{\tg x-\tg y}{1+\tg x\tg y} \\ \cotg( x+ y) & = & \frac{\cotg y+\cotg x}{\cotg x\cotg y-1} \\ \cotg( x- y) & = & \frac{\cotg y-\cotg x}{\cotg x\cotg y+1} \end{eqnarray}
\begin{eqnarray} \sin^2x + \cos^2x & = & \sin x\sin x + \cos x\cos x \;\; \stackrel{(\ref{cm})}{=} \;\; \cos(x-x) \;\; = \;\; \cos0 \;\; = \;\; 1 \label{pv} \\ & & \hfil \text{(Pythagorova věta)} \nonumber \\ \end{eqnarray} \begin{eqnarray} \sin(2x)& = & \sin(x+x) \;\; \stackrel{(\ref{sp})}{=} \;\; \sin x\cos x+\sin x\cos x \;\; = \;\; 2\sin x\cos x \\ \cos(2x)& = & \cos(x+x) \;\; \stackrel{(\ref{cp})}{=} \;\; \cos x\cos x - \sin x\sin x \;\; = \;\; \cos^2 x - \sin^2 x \label{c2x} \\ \end{eqnarray} \begin{eqnarray} \sin x & \stackrel{(\ref{pv})}{=} & \sqrt{1-\cos^2 x},\quad\text{pro}\ x\in[0,\pi] \\ \cos x & \stackrel{(\ref{pv})}{=} & \sqrt{1-\sin^2 x},\quad\text{pro}\ x\in\left[ -\frac{\pi}{2},\frac{\pi}{2}\right] \\ \end{eqnarray} \begin{eqnarray} \cos(2x) & \stackrel{(\ref{c2x})}{=} & \cos^2 x - \sin^2 x \;\; \stackrel{(\ref{pv})}{=} \;\; (1-\sin^2 x)-\sin^2 x \;\; = \;\; 1-2\sin^2 x \label{c2xs} \\ \cos(2x) & \stackrel{(\ref{c2x})}{=} & \cos^2 x - \sin^2 x \;\; \stackrel{(\ref{pv})}{=} \;\; \cos^2 x - (1-\cos^2 x) \;\; = \;\; 2\cos^2 x-1 \label{c2xc}\\ \end{eqnarray} \begin{eqnarray} \sin^2 x & \stackrel{(\ref{c2xs})}{=} & \frac{1-\cos(2 x)}{2} \\ \cos^2 x & \stackrel{(\ref{c2xc})}{=} & \frac{1+\cos(2 x)}{2} \end{eqnarray}
| radiány | $0$ | $\frac{\pi}{6}$ | $\frac{\pi}{4}$ | $\frac{\pi}{3}$ | $\frac{\pi}{2}$ | $\frac{2}{3}\pi$ | $\frac{3}{4}\pi$ | $\frac{5}{6}\pi$ | $\pi$ | $\frac{7}{6}\pi$ | $\frac{5}{4}\pi$ | $\frac{4}{3}\pi$ | $\frac{3}{2}\pi$ | $\frac{5}{3}\pi$ | $\frac{7}{4}\pi$ | $\frac{11}{6}\pi$ |
| stupně | $0^{\circ}$ | $30^{\circ}$ | $45^{\circ}$ | $60^{\circ}$ | $90^{\circ}$ | $120^{\circ}$ | $135^{\circ}$ | $150^{\circ}$ | $180^{\circ}$ | $210^{\circ}$ | $225^{\circ}$ | $240^{\circ}$ | $270^{\circ}$ | $300^{\circ}$ | $315^{\circ}$ | $330^{\circ}$ |
| $\sin x$ | $0$ | $\frac{1}{2}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{3}}{2}$ | $1$ | $\frac{\sqrt{3}}{2}$ | $\frac{\sqrt{2}}{2}$ | $\frac{1}{2}$ | $0$ | $-\frac{1}{2}$ | $-\frac{\sqrt{2}}{2}$ | $-\frac{\sqrt{3}}{2}$ | $-1$ | $-\frac{\sqrt{3}}{2}$ | $-\frac{\sqrt{2}}{2}$ | $-\frac{1}{2}$ |
| $\cos x$ | $1$ | $\frac{\sqrt{3}}{2}$ | $\frac{\sqrt{2}}{2}$ | $\frac{1}{2}$ | $0$ | $-\frac{1}{2}$ | $-\frac{\sqrt{2}}{2}$ | $-\frac{\sqrt{3}}{2}$ | $-1$ | $-\frac{\sqrt{3}}{2}$ | $-\frac{\sqrt{2}}{2}$ | $-\frac{1}{2}$ | $0$ | $\frac{1}{2}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{3}}{2}$ |
| $\tg x$ | $0$ | $\frac{\sqrt{3}}{3}$ | $1$ | $\sqrt{3}$ | $\pm\infty$ | $-\sqrt{3}$ | $-1$ | $-\frac{\sqrt{3}}{3}$ | $0$ | $\frac{\sqrt{3}}{3}$ | $1$ | $\sqrt{3}$ | $\pm\infty$ | $-\sqrt{3}$ | $-1$ | $-\frac{\sqrt{3}}{3}$ |
| $\cotg x$ | $\pm\infty$ | $\sqrt{3}$ | $1$ | $\frac{\sqrt{3}}{3}$ | $0$ | $-\frac{\sqrt{3}}{3}$ | $-1$ | $-\sqrt{3}$ | $\pm\infty$ | $\sqrt{3}$ | $1$ | $\frac{\sqrt{3}}{3}$ | $0$ | $-\frac{\sqrt{3}}{3}$ | $-1$ | $-\sqrt{3}$ |
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