Research website of Vyacheslav Gorchilin
All formulas
Formula. Math. Trigonometry
The sum or difference of functions
$\sin \alpha \pm \sin \beta$
$\cos \alpha + \cos \beta$
$\cos \alpha - \cos \beta$
$\mathtt{tg} \alpha \pm \mathtt{tg} \beta$
$\mathtt{ctg} \alpha \pm \mathtt{ctg} \beta$
$\sin 2\alpha + \sin 2\beta + \sin 2\gamma$
$\sin 2 \alpha$
$\cos 2 \alpha$
$\mathtt{tg} 2 \alpha$
$\mathtt{ctg} 2 \alpha$
$\sin 3\alpha$
$\cos 3\alpha$
$\mathtt{tg} 3\alpha$
$\mathtt{ctg} 3\alpha$
$\sin {\biggl (}{\alpha \over 2}{\biggr )}$
$\cos {\biggl (}{\alpha \over 2}{\biggr )}$
$\mathtt{tg}{\biggl (}{\alpha \over 2}{\biggr)}$
The sum (difference) of sines $\sin \alpha \pm \sin \beta = 2 \sin \frac{\alpha \pm \beta}{2} \cos \frac{\alpha \mp \beta}{2}$
The sum of the cosines $\cos \alpha + \cos \beta = 2 \cos \frac{\alpha + \beta}{2} \cos \frac{\alpha - \beta}{2}$
The difference of the cosines $\cos \alpha - \cos \beta = - 2 \sin \frac{\alpha + \beta}{2} \sin \frac{\alpha - \beta}{2}$
The sum of the tangents $\mathtt{tg} \alpha \pm \mathtt{tg} \beta = \frac{\sin (\alpha \pm \beta)}{\cos \alpha \cos \beta}$
The difference between the cotangent $\mathtt{ctg} \alpha \pm \mathtt{ctg} \beta = \frac{\sin (\beta \pm \alpha)}{\sin \alpha \sin \beta}$
The product or sum of three sines $\sin 2\alpha + \sin 2\beta + \sin 2\gamma = 4 \sin\alpha \sin\beta \sin\gamma$
The sine double angle $\sin 2 \alpha = 2 \sin\alpha \, \cos\alpha$
The cosine double angle $\cos 2 \alpha = \cos^2 \alpha - \sin^2 \alpha = 2 \cos^2 \alpha - 1 = 1 - 2 \sin^2 \alpha$
Tangent double angle $\mathtt{tg} 2 \alpha = \frac{2\,\mathtt{tg} \alpha}{1 - \mathtt{tg}^2 \alpha}$
Cotangent double angle $\mathtt{ctg} 2 \alpha = \frac{\mathtt{ctg}^2 \alpha - 1}{2\,\mathtt{ctg} \alpha}$
The sine of a triple angle $\sin 3\alpha = 3 \sin \alpha - 4 \sin^3\alpha$
The cosine triple angle $\cos 3\alpha = 4 \cos^3\alpha - 3 \cos \alpha$
The triple tangent of the angle $\mathtt{tg} 3\alpha = \frac{3\,\mathtt{tg}\alpha - \mathtt{tg}^3\alpha}{1 - 3\,\mathtt{tg}^2\alpha}$
The cotangent triple angle $\mathtt{ctg} 3\alpha = \frac{3\,\mathtt{ctg}\alpha - \mathtt{ctg}^3\alpha}{1 - 3\,\mathtt{ctg}^2\alpha}$
The sine of the half angle $\sin {\biggl (}{\alpha \over 2}{\biggr )} = \pm {\sqrt{1-\cos \alpha \over 2}}$
The cosine of the half angle $\cos {\biggl (}{\alpha \over 2}{\biggr )} = \pm {\sqrt{1+\cos \alpha \over 2}}$
The tangent of the half angle $\mathtt{tg}{\biggl (}{\alpha \over 2}{\biggr)} = \pm {\sqrt{1-\cos \alpha \over 1+\cos \alpha}} = {\sin \alpha \over 1+\cos \alpha} = {1-\cos \alpha \over \sin \alpha}$