Cosine of Angle plus Straight Angle

From ProofWiki
Jump to navigation Jump to search

Theorem

$\map \cos {x + \pi} = -\cos x$


Proof 1

\(\displaystyle \cos \paren {x + \pi}\) \(=\) \(\displaystyle \cos x \cos \pi - \sin x \sin \pi\) Cosine of Sum
\(\displaystyle \) \(=\) \(\displaystyle \cos x \cdot \paren {-1} - \sin x \cdot 0\) Cosine of Straight Angle and Sine of Straight Angle
\(\displaystyle \) \(=\) \(\displaystyle -\cos x\)

$\blacksquare$


Proof 2

\(\displaystyle \map \cos {x + \pi}\) \(=\) \(\displaystyle \map \Re {\map \cos {x + \pi} + i \, \map \sin {x + \pi} }\)
\(\displaystyle \) \(=\) \(\displaystyle \map \Re {e^{i \paren {x + \pi} } }\) Euler's Formula
\(\displaystyle \) \(=\) \(\displaystyle \map \Re {e^{i x + i \pi} }\)
\(\displaystyle \) \(=\) \(\displaystyle \map \Re {e^{i x} e^{i \pi} }\) Exponential of Sum: Complex Numbers
\(\displaystyle \) \(=\) \(\displaystyle \map \Re {-e^{i x} }\) Euler's Identity
\(\displaystyle \) \(=\) \(\displaystyle -\map \Re {e^{i x} }\)
\(\displaystyle \) \(=\) \(\displaystyle -\map \Re {\cos x + i \cos x}\) Euler's Formula
\(\displaystyle \) \(=\) \(\displaystyle -\cos x\)

$\blacksquare$


Proof 3

\(\displaystyle \map \cos {x + \pi}\) \(=\) \(\displaystyle \frac 1 2 \paren {e^{i \paren {x + \pi} } + e^{-i \paren {x + \pi} } }\) Cosine Exponential Formulation
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 \paren {e^{i x} e^{i \pi} + e^{-i x} e^{-i \pi} }\) Exponential of Sum: Complex Numbers
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 \paren {-e^{i x} - e^{-i x} }\) Euler's Identity
\(\displaystyle \) \(=\) \(\displaystyle -\frac 1 2 \paren {e^{i x} + e^{-i x} }\)
\(\displaystyle \) \(=\) \(\displaystyle -\cos x\) Cosine Exponential Formulation

$\blacksquare$


Also see


Sources