Sine of Difference

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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 1

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

$\blacksquare$


Proof 2

\(\ds \map \cos {90 \degrees + a - b}\) \(=\) \(\ds \map \cos {90 \degrees + a} \cos b + \map \sin {90 \degrees + a} \sin b\) Cosine of Difference
\(\ds \leadsto \ \ \) \(\ds \map \sin {a - b}\) \(=\) \(\ds \sin a \cos b - \cos a \sin b\) Cosine of Angle plus Right Angle, Sine of Angle plus Right Angle

$\blacksquare$


Also see


Historical Note

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


Sources

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