Sine plus Sine/Proof 1
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Theorem
- $\sin \alpha + \sin \beta = 2 \map \sin {\dfrac {\alpha + \beta} 2} \map \cos {\dfrac {\alpha - \beta} 2}$
Proof
\(\text {(1)}: \quad\) | \(\ds \map \sin {A + B}\) | \(=\) | \(\ds \sin A \cos B + \cos A \sin B\) | Sine of Sum | ||||||||||
\(\text {(2)}: \quad\) | \(\ds \map \sin {A - B}\) | \(=\) | \(\ds \sin A \cos B - \cos A \sin B\) | Sine of Difference | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \sin {A + B} + \map \sin {A - B}\) | \(=\) | \(\ds 2 \sin A \cos B\) | adding $(1)$ and $(2)$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sin \alpha + \sin \beta\) | \(=\) | \(\ds 2 \map \sin {\dfrac {\alpha + \beta} 2} \map \cos {\dfrac {\alpha - \beta} 2}\) | setting $A + B = \alpha$ and $A - B = \beta$ |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: The product formulae