Cosine of Angle plus Straight Angle/Proof 4
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Theorem
- $\map \cos {x + \pi} = -\cos x$
Proof
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:
- $\map \cos {x + \pi} = -\map \sin {x + \dfrac \pi 2} = -\cos x$
$\blacksquare$