Formulas and Tables

Trigonometry

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
    \({\tan^3}\alpha = {\large\frac{{{{\sin }^3}\alpha }}{{{{\cos }^3}\alpha}}\normalsize} =\) \( {\large\frac{{3\sin \alpha – \sin 3\alpha }}{{3\cos \alpha + \cos 3\alpha}}\normalsize}\)