Formulas and Tables

Trigonometry

Sum-to-Product Identities

Angles (arguments of functions): \(\alpha\), \(\beta\)

Trigonometric functions: \(\sin \alpha,\) \(\cos \alpha,\) \(\tan \alpha,\) \(\cot \alpha\)

  1. Sum of sines
    \(\sin \alpha + \sin \beta =\) \( 2\sin \large\frac{{\alpha + \beta }}{2}\normalsize \cos \large\frac{{\alpha – \beta }}{2}\normalsize\)
  2. Difference of sines
    \(\sin \alpha – \sin \beta =\) \( 2\cos \large\frac{{\alpha + \beta }}{2}\normalsize \sin \large\frac{{\alpha – \beta }}{2}\normalsize\)
  3. Sum of cosines
    \(\cos \alpha + \cos \beta =\) \( 2\cos \large\frac{{\alpha + \beta }}{2}\normalsize \cos \large\frac{{\alpha – \beta }}{2}\normalsize\)
  4. Difference of cosines
    \(\cos \alpha – \cos \beta =\) \( -2\sin \large\frac{{\alpha + \beta }}{2}\normalsize \sin \large\frac{{\alpha – \beta }}{2}\normalsize\)
  5. Sum of tangents
    \(\tan\alpha + \tan \beta =\) \( \large\frac{{\sin \left( {\alpha + \beta } \right)}}{{\cos \alpha \cdot \cos \beta }}\normalsize\)
  6. Difference of tangents
    \(\tan\alpha – \tan \beta =\) \( \large\frac{{\sin \left( {\alpha – \beta } \right)}}{{\cos \alpha \cdot \cos \beta }}\normalsize\)
  7. Sum of cotangents
    \(\cot\alpha + \cot \beta =\) \( \large\frac{{\sin \left( {\beta + \alpha } \right)}}{{\sin \alpha \cdot \sin \beta }}\normalsize\)
  8. Difference of cotangents
    \(\cot\alpha – \cot \beta =\) \( \large\frac{{\sin \left( {\beta – \alpha } \right)}}{{\sin \alpha \cdot \sin \beta }}\normalsize\)
  9. Sum of cosine and sine
    \(\cos\alpha + \sin \alpha =\) \( \sqrt 2 \cos \left( {{\large\frac{\pi }{4}\normalsize} – \alpha } \right) =\) \( \sqrt 2 \sin\left( {{\large\frac{\pi }{4}\normalsize} + \alpha } \right)\)
  10. Difference of cosine and sine
    \(\cos\alpha – \sin \alpha =\) \( \sqrt 2 \sin \left( {{\large\frac{\pi }{4}\normalsize} – \alpha } \right) =\) \( \sqrt 2 \cos\left( {{\large\frac{\pi }{4}\normalsize} + \alpha } \right)\)
  11. Sum of tangent and cotangent
    \(\tan\alpha + \cot \beta =\) \( \large\frac{{\cos \left( {\alpha – \beta } \right)}}{{\cos \alpha \cdot \sin \beta }}\normalsize\)
  12. Difference of tangent and cotangent
    \(\tan\alpha – \cot \beta =\) \( -\large\frac{{\cos \left( {\alpha + \beta } \right)}}{{\cos \alpha \cdot \sin \beta }}\normalsize\)
  13. \(1 + \cos \alpha = 2\,{\cos ^2}\large\frac{\alpha }{2}\normalsize\)
  14. \(1 – \cos \alpha = 2\,{\sin ^2}\large\frac{\alpha }{2}\normalsize\)
  15. \(1 + \sin \alpha =\) \( 2\,{\cos ^2}\left( {\large\frac{\pi }{4} – \frac{\alpha }{2}\normalsize} \right)\)
  16. \(1 – \sin \alpha =\) \( 2\,{\sin ^2}\left( {\large\frac{\pi }{4} – \frac{\alpha }{2}\normalsize} \right)\)