$$
\begin{aligned}
& \sin\alpha = \frac{a}{c} \ \ \ && \sin\beta = \frac{b}{c} \\ \\
& \cos\alpha = \frac{b}{c} && \cos\beta = \frac{a}{c} \\ \\
& \tan\alpha = \frac{a}{b} && \tan\beta = \frac{b}{a} \\ \\
& \cot\alpha = \frac{b}{a} && \cot\beta = \frac{a}{b}
\end{aligned}
$$
$$
\begin{aligned}
&\sin^2\alpha + \cos^2\alpha = 1 \\ \\
&\tan\alpha \cdot \cot\alpha = 1 \ \Rightarrow \\ \\ &\cot\alpha = \frac{1}{\tan\alpha} \\ \\
&\tan\alpha = \frac{\sin\alpha}{\cos\alpha} \\ \\
&\cot\alpha = \frac{\cos\alpha}{\sin\alpha}
\end{aligned}
$$
$$
\begin{aligned}
&\sin(\alpha + \beta) = \\& = \sin\alpha\cos\beta + \cos\alpha\sin\beta \\ \\
&\sin(\alpha - \beta) = \\& = \sin\alpha\cos\beta - \cos\alpha\sin\beta \\ \\
&\cos(\alpha + \beta) = \\& = \cos\alpha\cos\beta - \sin\alpha\sin\beta \\ \\
&\cos(\alpha - \beta) = \\& = \cos\alpha\cos\beta + \sin\alpha\sin\beta \\ \\
&\tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta} \\ \\
&\tan(\alpha - \beta) = \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha\tan\beta}
\end{aligned}
$$
$$
\begin{aligned}
&\sin\alpha + \sin\beta = \\& = 2\cdot\sin\frac{\alpha + \beta}{2}\cdot\cos\frac{\alpha - \beta}{2} \\ \\
&\sin\alpha - \sin\beta = \\& = 2\cdot\cos\frac{\alpha + \beta}{2}\cdot\sin\frac{\alpha - \beta}{2} \\ \\
&\cos\alpha + \cos\beta = \\& = 2\cdot\cos\frac{\alpha + \beta}{2}\cdot\cos\frac{\alpha - \beta}{2} \\ \\
&\cos\alpha - \cos\beta = \\& = -2\cdot\sin\frac{\alpha + \beta}{2}\cdot\sin\frac{\alpha - \beta}{2}
\end{aligned}
$$
$$
\begin{aligned}
& \sin 2\alpha = 2\cdot\sin\alpha\cos\alpha \\ \\
& \cos 2\alpha = \cos^2\alpha - \sin^2\alpha \\ \\
& \tan 2\alpha = \frac{2\cdot\tan\alpha}{1 - {\tan}^2\alpha}
\end{aligned}
$$
$$
\begin{aligned}
& \left|\sin\frac{\alpha}{2}\right| = \sqrt{\frac{1-\cos\alpha}{2}} \\ \\
& \left|\cos\frac{\alpha}{2}\right| = \sqrt{\frac{1+\cos\alpha}{2}} \\ \\
& \left|\tan\frac{\alpha}{2}\right| = \sqrt{\frac{1-\cos\alpha}{1+\cos\alpha}}
\end{aligned}
$$
$$
\begin{aligned}
& \sin(-\alpha) = -\sin\alpha \\ \\
& \cos(-\alpha) = \cos\alpha \\ \\
& \tan(-\alpha) = -\tan\alpha \\ \\
& \cot(-\alpha) = -\cot\alpha
\end{aligned}
$$
