Formulas and Tables

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