Formulas

Trig Identities

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Addition and Subtraction Formulas

  • Angles: \(\alpha\), \(\beta\)
    Trigonometric functions: \(\sin \alpha,\) \(\cos \alpha,\) \(\tan \alpha,\) \(\cot \alpha\)
    1. Sine addition formula
      \(\sin \left( {\alpha + \beta } \right) =\) \( \sin \alpha \cos \beta \,+\) \( \cos \alpha \sin \beta \)
    2. Sine subtraction formula
      \(\sin \left( {\alpha – \beta } \right) =\) \( \sin \alpha \cos \beta \,-\) \( \cos \alpha \sin \beta \)
    3. Cosine addition formula
      \(\cos \left( {\alpha + \beta } \right) =\) \( \cos \alpha \cos \beta \,-\) \( \sin \alpha \sin \beta \)
    4. Cosine subtraction formula
      \(\cos \left( {\alpha – \beta } \right) =\) \( \cos \alpha \cos \beta \,+\) \( \sin \alpha \sin \beta \)
    5. Tangent addition formula
      \(\tan \left( {\alpha + \beta } \right) = \large\frac{{\tan \alpha + \tan \beta }}{{1 – \tan \alpha \tan \beta }}\normalsize\)
    6. Tangent subtraction formula
      \(\tan \left( {\alpha – \beta } \right) = \large\frac{{\tan \alpha – \tan \beta }}{{1 + \tan \alpha \tan \beta }}\normalsize\)
    7. Cotangent addition formula
      \(\cot \left( {\alpha + \beta } \right) = \large\frac{{1 – \tan \alpha \tan \beta }}{{\tan \alpha + \tan \beta }}\normalsize\)
    8. Cotangent subtraction formula
      \(\cot \left( {\alpha – \beta } \right) = \large\frac{{1 + \tan \alpha \tan \beta }}{{\tan \alpha – \tan \beta }}\normalsize\)