Formulas

Trig Identities

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Powers of Trigonometric Functions

  • Angle: \(\alpha\)
    Trigonometric functions: \(\sin \alpha\), \(\cos \alpha\), \(\tan \alpha\)
    1. Sine squared
      \({\sin^2}\alpha =\) \( \large\frac{{1 – \cos 2\alpha }}{2}\normalsize\)
    2. Sine cubed
      \({\sin^3}\alpha =\) \( \large\frac{{3\sin \alpha – \sin 3\alpha }}{4}\normalsize\)
    3. Sine to the fourth power
      \({\sin^4}\alpha =\) \( \large\frac{{\cos 4\alpha – 4\cos 2\alpha + 3}}{8}\normalsize\)
    4. Sine to the fifth power
      \({\sin^5}\alpha =\) \( \large\frac{{10\sin \alpha – 5\sin 3\alpha + \sin 5\alpha}}{16}\normalsize\)
    5. Sine to the sixth power
      \({\sin^6}\alpha =\) \( \large\frac{{10 – 15\cos 2\alpha + 6\cos 4\alpha – \cos 6\alpha}}{32}\normalsize\)
    6. Cosine squared
      \({\cos^2}\alpha =\) \( \large\frac{{1 + \cos 2\alpha }}{2}\normalsize\)
    7. Cosine cubed
      \({\cos^3}\alpha =\) \( \large\frac{{3\cos \alpha + \cos 3\alpha }}{4}\normalsize\)
    8. Cosine to the fourth power
      \({\cos^4}\alpha =\) \( \large\frac{{\cos 4\alpha + 4\cos 2\alpha + 3}}{8}\normalsize\)
    9. Cosine to the fifth power
      \({\cos^5}\alpha =\) \( \large\frac{{10\cos \alpha + 5\sin 3\alpha + \cos 5\alpha}}{16}\normalsize\)
    10. Cosine to the sixth power
      \({\cos^6}\alpha =\) \( \large\frac{{10 + 15\cos 2\alpha + 6\cos 4\alpha + \cos 6\alpha}}{32}\normalsize\)
    11. Tangent squared
      \({\tan^2}\alpha = {\large\frac{{{{\sin }^2}\alpha }}{{{{\cos }^2}\alpha}}\normalsize} =\) \( {\large\frac{{1 – \cos 2\alpha }}{{1 + \cos 2\alpha }}\normalsize}\)
    12. Tangent cubed
      \({\tan^3}\alpha = {\large\frac{{{{\sin }^3}\alpha }}{{{{\cos }^3}\alpha}}\normalsize} =\) \( {\large\frac{{3\sin \alpha – \sin 3\alpha }}{{3\cos \alpha + \cos 3\alpha}}\normalsize}\)