Formulas

Trig Identities

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Double and Multiple Angle Formulas

  • Angles (arguments of functions): \(\alpha\)
    Trigonometric functions: \(\sin \alpha,\) \(\cos \alpha,\) \(\tan \alpha,\) \(\cot \alpha\)
    1. Sine of a double angle
      \(\sin 2\alpha = 2\sin \alpha \cos \alpha \)
    2. Cosine of a double angle
      \(\cos 2\alpha = {\cos ^2}\alpha – {\sin ^2}\alpha =\) \( 1 – 2\,{\sin ^2}\alpha =\) \( 2\,{\cos ^2}\alpha – 1\)
    3. Tangent of a double angle
      \(\tan 2\alpha = {\large\frac{{2\tan \alpha }}{{1 – {{\tan }^2}\alpha}}\normalsize} =\) \({\large\frac{2}{{\cot \alpha – \tan \alpha }}\normalsize}\)
    4. Cotangent of a double angle
      \(\cot 2\alpha = {\large\frac{{{{\cot }^2}\alpha – 1}}{{2\cot \alpha}}\normalsize} =\) \({\large\frac{{\cot \alpha – \tan \alpha }}{2}\normalsize}\)
    5. Sine of a triple angle
      \({\sin 3\alpha = 3\sin\alpha – 4\,{\sin^3} \alpha }=\) \( 3\,{\cos ^2}\alpha \sin \alpha – {\sin ^3}\alpha \)
    6. \(\sin 4\alpha = 4\sin \alpha \cos \alpha \,-\) \(8\,{\sin ^3}\alpha \cos \alpha \)
    7. \(\sin 5\alpha = 5\sin\alpha \,-\) \(20\,{\sin ^3}\alpha \,+\) \(16\,{\sin ^5}\alpha \)
    8. Cosine of a triple angle
      \(\cos 3\alpha =\) \( 4\,{\cos^3} \alpha – 3\cos \alpha =\) \( {\cos ^3}\alpha – 3\,{\sin ^2}\alpha \cos \alpha\)
    9. \(\cos 4\alpha = 8\,{\cos ^4}\alpha \,-\) \(8\,{\cos ^2}\alpha + 1\)
    10. \(\cos 5\alpha = 16\,{\cos ^5}\alpha \,-\) \(20\,{\cos ^3}\alpha \,+\) \(5\cos \alpha \)
    11. Tangent of a triple angle
      \(\tan 3\alpha =\) \(\large\frac{{3\tan \alpha – {{\tan }^3}\alpha }}{{1 – 3\,{{\tan }^2}\alpha }}\normalsize\)
    12. \(\tan 4\alpha =\) \(\large\frac{{4\tan \alpha – 4\,{{\tan }^3}\alpha }}{{1 – 6\,{{\tan }^2}\alpha + \,{{\tan }^4}\alpha }}\normalsize\)
    13. \(\tan 5\alpha =\) \(\large\frac{{\,{{\tan }^5}\alpha – 10\,{{\tan }^3}\alpha + 5\tan \alpha }}{{1 – 10\,{{\tan }^2}\alpha + 5\,{{\tan }^4}\alpha }}\normalsize\)
    14. Cotangent of a triple angle
      \(\cot 3\alpha =\) \( \large\frac{{\,{\cot^3}\alpha – 3\cot \alpha }}{{3\,{{\cot }^2}\alpha – 1}}\normalsize\)
    15. \(\cot 4\alpha =\) \( \large\frac{{1 – 6\,{{\tan }^2}\alpha + {{\tan }^4}\alpha }}{{4\tan \alpha – 4\,{{\tan }^3}\alpha}}\normalsize\)
    16. \(\cot 5\alpha =\) \( \large\frac{{1 – 10\,{{\tan }^2}\alpha + 5\,{{\tan }^4}\alpha }}{{\,{{\tan }^5}\alpha – 10\,{{\tan }^3}\alpha + 5\tan \alpha }}\normalsize\)