Cosine of Complement equals Sine/Proof 3
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Theorem
- $\map \cos {\dfrac \pi 2 - \theta} = \sin \theta$
Proof
| \(\ds \map \cos {\dfrac \pi 2 - \theta}\) | \(=\) | \(\ds \frac 1 2 \paren {e^{i \paren {\frac \pi 2 - \theta} } + e^{-i \paren {\frac \pi 2 - \theta} } }\) | Euler's Cosine Identity | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {e^{i \frac \pi 2} e^{-i \theta} + e^{-i \frac \pi 2} e^{i \theta} }\) | Exponential of Sum: Complex Numbers | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {\paren {\map \cos {\frac \pi 2} + i \, \map \sin {\frac \pi 2} } e^{-i \theta} + \paren {\map \cos {-\frac \pi 2} + i \, \map \sin {-\frac \pi 2} } e^{i \theta} }\) | Euler's Formula | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {i e^{-i \theta} - i e^{i \theta} }\) | Cosine of Right Angle, Sine of Right Angle, Cosine Function is Even, Sine Function is Odd | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {2 i} \paren {e^{i \theta} - e^{-i \theta} }\) | Definition of Imaginary Unit | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sin \theta\) | Euler's Sine Identity |
$\blacksquare$