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Advanced Trigonometry Identities

This page shows some advanced trig identities that is more common in math competition.

Product-to-Sum Formulas​

19​

sin⁡αcos⁡β=sin⁡(α+β)+sin⁡(α−β)2\sin{\alpha}\cos{\beta}=\frac{\sin{(\alpha+\beta)}+\sin{(\alpha-\beta)}}{2}
Derivation

20​

cos⁡αsin⁡β=sin⁡(α+β)−sin⁡(α−β)2\cos{\alpha}\sin{\beta}=\frac{\sin{(\alpha+\beta)}-\sin{(\alpha-\beta)}}{2}
Derivation

21​

cos⁡αcos⁡β=cos⁡(α+β)+cos⁡(α−β)2\cos{\alpha}\cos{\beta}=\frac{\cos{(\alpha+\beta)}+\cos{(\alpha-\beta)}}{2}
Derivation

22​

sin⁡αsin⁡β=−cos⁡(α+β)+cos⁡(α−β)2\sin{\alpha}\sin{\beta}=\frac{-\cos{(\alpha+\beta)}+\cos{(\alpha-\beta)}}{2}
Derivation

Sum-to-Product Formulas​

23​

sin⁡α+sin⁡β=2sin⁡α+β2cos⁡α−β2\sin{\alpha}+\sin{\beta}=2\sin{\frac{\alpha+\beta}{2}}\cos{\frac{\alpha-\beta}{2}}
Derivation

24​

sin⁡α−sin⁡β=2cos⁡α+β2sin⁡α−β2\sin{\alpha}-\sin{\beta}=2\cos{\frac{\alpha+\beta}{2}}\sin{\frac{\alpha-\beta}{2}}
Derivation

25​

cos⁡α+cos⁡β=2cos⁡α+β2cos⁡α−β2\cos{\alpha}+\cos{\beta}=2\cos{\frac{\alpha+\beta}{2}}\cos{\frac{\alpha-\beta}{2}}
Derivation

26​

cos⁡α−cos⁡β=−2sin⁡α+β2sin⁡α−β2\cos{\alpha}-\cos{\beta}=-2\sin{\frac{\alpha+\beta}{2}}\sin{\frac{\alpha-\beta}{2}}
Derivation

Auxiliary Angle Formula​

27​

Asin⁡α+Bcos⁡α=A2+B2sin⁡α+arctan⁡BAA\sin{\alpha}+B\cos{\alpha}=\sqrt{A^2+B^2}\sin{\alpha+\arctan{\frac{B}{A}}}
Derivation