Angles (arguments of functions): \(\alpha\)

Trigonometric functions: \(\sin \alpha,\) \(\cos \alpha,\) \(\tan \alpha,\) \(\cot \alpha\)

- Sine of a half angle

\(\sin {\large\frac{\alpha }{2}\normalsize} = \pm \sqrt {\large\frac{{1 – \cos \alpha }}{2}\normalsize}\)

Note: The sign in front of the radical is chosen depending on the quadrant in which the angle \(\large{\frac{\alpha}{2}}\normalsize\) on the left side lies. This rule also applies to formulas \(2-4.\) - Cosine of a half angle

\(\cos {\large\frac{\alpha }{2}\normalsize} = \pm \sqrt {\large\frac{{1 + \cos \alpha }}{2}\normalsize} \) - Tangent of a half angle

\(\tan {\large\frac{\alpha }{2}\normalsize} = \pm \sqrt {\large\frac{{1 – \cos \alpha }\normalsize}{{1 + \cos \alpha }}} =\) \({\large\frac{{\sin \alpha }}{{1 + \cos \alpha }}\normalsize} =\) \({\large\frac{{1 – \cos \alpha }}{{\sin \alpha }}\normalsize} =\) \( \csc \alpha – \cot \alpha \) - Cotangent of a half angle

\(\cot {\large\frac{\alpha }{2}\normalsize} = \pm \sqrt {\large\frac{{1 + \cos \alpha }\normalsize}{{1 – \cos \alpha }}} =\) \({\large\frac{{\sin \alpha }}{{1 – \cos \alpha }}\normalsize} =\) \({\large\frac{{1 + \cos \alpha }}{{\sin \alpha }}\normalsize} =\) \( \csc \alpha + \cot \alpha \) - Tangent half angle formula for sine

\(\sin\alpha = \large\frac{{2\tan \frac{\alpha }{2}}}{{1 + {{\tan }^2}\frac{\alpha }{2}}}\normalsize\) - Tangent half angle formula for cosine

\(\cos\alpha = \large\frac{{1 – {{\tan }^2}\frac{\alpha }{2}}}{{1 + {{\tan }^2}\frac{\alpha }{2}}}\normalsize\) - Tangent half angle formula for tangent

\(\tan\alpha = \large\frac{{2\tan \frac{\alpha }{2}}}{{1 – {{\tan }^2}\frac{\alpha }{2}}}\normalsize\) - Tangent half angle formula for cotangent

\(\cot\alpha = \large\frac{{1 – {{\tan }^2}\frac{\alpha }{2}}}{{2\tan \frac{\alpha }{2}}}\normalsize\)