Sine of Sum/Proof 3
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Theorem
- $\map \sin {a + b} = \sin a \cos b + \cos a \sin b$
Proof
| \(\ds \sin a \cos b + \cos a \sin b\) | \(=\) | \(\ds \paren {\frac {e^{i a} - e^{-i a} }{2 i} } \cos b + \cos a \paren {\frac {e^{i b} - e^{-i b} }{2 i} }\) | Euler's Sine Identity | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\frac {e^{i a} - e^{-i a} } {2 i} } \paren {\frac {e^{i b} + e^{-i b} } 2}
+ \paren {\frac {e^{i a} + e^{-i a} } 2} \paren {\frac {e^{i b} - e^{-i b} } {2 i} }\)
|
Euler's Cosine Identity | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^{i a} e^{i b} + e^{i a} e^{-i b} - e^{-i a} e^{i b} - e^{-i a} e^{-i b} } {4 i}\) | expanding | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac {e^{i a} e^{i b} - e^{i a} e^{-i b} + e^{-i a} e^{i b} - e^{-i a} e^{-i b} } {4 i}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^{i a} e^{i b} - e^{-i a} e^{-i b} } {2 i}\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^{i \paren {a + b} } - e^{-i \paren {a + b} } } {2 i}\) | Exponential of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \sin {a + b}\) | Euler's Sine Identity |
$\blacksquare$