Sine of Complement equals Cosine/Proof 3
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Theorem
- $\map \sin {\dfrac \pi 2 - \theta} = \cos \theta$
Proof
| \(\ds \map \sin {\dfrac \pi 2 - \theta}\) | \(=\) | \(\ds \map \Im {e^{i \paren {\frac \pi 2 - \theta} } }\) | Euler's Formula | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \Im {e^{i \frac \pi 2} e^{-i \theta} }\) | Exponent Combination Laws | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \Im {\paren {\cos \dfrac \pi 2+i \sin \dfrac \pi 2} e^{-i \theta} }\) | Euler's Formula | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \Im {i e^{-i \theta} }\) | Cosine of Right Angle, Sine of Right Angle | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \Re {e^{-i \theta} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \map \cos {-\theta}\) | Euler's Formula | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos \theta\) | Cosine Function is Even |
$\blacksquare$
 | This article, or a section of it, needs explaining. In particular: Just had a thought -- do we need a result which says something like $\Im i z = \map \Re z$? It's not completely obvious, specially since $\Re i z = -\Im z$. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |