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Trigonometry

Half-Angle Formulas

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

Trigonometric functions: $$\sin \alpha,$$ $$\cos \alpha,$$ $$\tan \alpha,$$ $$\cot \alpha$$

1. 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 $$2-4.$$
2. Cosine of a half angle
$$\cos {\large\frac{\alpha }{2}\normalsize} = \pm \sqrt {\large\frac{{1 + \cos \alpha }}{2}\normalsize}$$
3. Tangent of a half angle
$$\tan {\large\frac{\alpha }{2}\normalsize} = \pm \sqrt {\large\frac{{1 – \cos \alpha }\normalsize}{{1 + \cos \alpha }}} =$$ $${\large\frac{{\sin \alpha }}{{1 + \cos \alpha }}\normalsize} =$$ $${\large\frac{{1 – \cos \alpha }}{{\sin \alpha }}\normalsize} =$$ $$\csc \alpha – \cot \alpha$$
4. Cotangent of a half angle
$$\cot {\large\frac{\alpha }{2}\normalsize} = \pm \sqrt {\large\frac{{1 + \cos \alpha }\normalsize}{{1 – \cos \alpha }}} =$$ $${\large\frac{{\sin \alpha }}{{1 – \cos \alpha }}\normalsize} =$$ $${\large\frac{{1 + \cos \alpha }}{{\sin \alpha }}\normalsize} =$$ $$\csc \alpha + \cot \alpha$$
5. Tangent half angle formula for sine
$$\sin\alpha = \large\frac{{2\tan \frac{\alpha }{2}}}{{1 + {{\tan }^2}\frac{\alpha }{2}}}\normalsize$$
6. Tangent half angle formula for cosine
$$\cos\alpha = \large\frac{{1 – {{\tan }^2}\frac{\alpha }{2}}}{{1 + {{\tan }^2}\frac{\alpha }{2}}}\normalsize$$
7. Tangent half angle formula for tangent
$$\tan\alpha = \large\frac{{2\tan \frac{\alpha }{2}}}{{1 – {{\tan }^2}\frac{\alpha }{2}}}\normalsize$$
8. Tangent half angle formula for cotangent
$$\cot\alpha = \large\frac{{1 – {{\tan }^2}\frac{\alpha }{2}}}{{2\tan \frac{\alpha }{2}}}\normalsize$$