Cosine of Angle plus Straight Angle/Proof 2
Jump to navigation
Jump to search
Theorem
- $\map \cos {x + \pi} = -\cos x$
Proof
\(\ds \map \cos {x + \pi}\) | \(=\) | \(\ds \map \Re {\map \cos {x + \pi} + i \, \map \sin {x + \pi} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \Re {e^{i \paren {x + \pi} } }\) | Euler's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Re {e^{i x + i \pi} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \Re {e^{i x} e^{i \pi} }\) | Exponential of Sum: Complex Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Re {-e^{i x} }\) | Euler's Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds -\map \Re {e^{i x} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\map \Re {\cos x + i \cos x}\) | Euler's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds -\cos x\) |
$\blacksquare$