# Cosine of Angle plus Straight Angle It has been suggested that this page or section be merged into Sine and Cosine are Periodic on Reals/Corollary. (Discuss)

## 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

$\blacksquare$