Arcs differing by pi

Property — Arcs differing by $\pi$: $\pi + \alpha$

$\sin(\pi+\alpha) = -\sin\alpha,\qquad \cos(\pi+\alpha) = -\cos\alpha,\qquad \tan(\pi+\alpha) = \tan\alpha.$ Symmetry about the origin: the point $P_{\pi+\alpha}$ is diametrically opposite $P_\alpha$, so it has both coordinates with their signs reversed. The tangent, being the ratio of two quantities with reversed signs, stays unchanged: this is the reason why $\tan$ has period $\pi$.

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Topics: Trigonometry
Concepts: Associated arcs · Cosine · Sine · Tangent
Functions: Cosine · Sine · Tangent
Skills: Using formulae