Cosine of Angle plus Straight Angle

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Theorem

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


Proof 1

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

$\blacksquare$


Proof 2

\(\ds \map \cos {x + \pi}\) \(=\) \(\ds \map \Re {\map \cos {x + \pi} + i \, \map \sin {x + \pi} }\)
\(\ds \) \(=\) \(\ds \map \Re {e^{i \paren {x + \pi} } }\) 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
\(\ds \) \(=\) \(\ds -\cos x\)

$\blacksquare$


Proof 3

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

$\blacksquare$


Proof 4

From the discussion in the proof of Real Cosine Function is Periodic:

$\map \sin {x + \eta} = \cos x$
$\map \cos {x + \eta} = -\sin x$

for $\eta \in \R_{>0}$.

From Sine and Cosine are Periodic on Reals: Pi, we define $\pi \in \R$ as $\pi := 2 \eta$.

It follows that $\eta = \dfrac \pi 2$, thus:

$\map \cos {x + \pi} = -\map \sin {x + \dfrac \pi 2} = -\cos x$

$\blacksquare$


Also see


Sources

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