Sine of Sum/Corollary

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Corollary to Sine of Sum

$\map \sin {a - b} = \sin a \cos b - \cos a \sin b$

where $\sin$ denotes the sine and $\cos$ denotes the cosine.


Proof

\(\displaystyle \map \sin {a - b}\) \(=\) \(\displaystyle \sin a \, \map \cos {-b} + \cos a \, \map \sin {-b}\) Sine of Sum
\(\displaystyle \) \(=\) \(\displaystyle \sin a \cos b + \cos a \, \map \sin {-b}\) Cosine Function is Even
\(\displaystyle \) \(=\) \(\displaystyle \sin a \cos b - \cos a \sin b\) Sine Function is Odd

$\blacksquare$


Historical Note

The Sine of Sum formula and its corollary were proved by François Viète in about $1579$.


Sources