Periodicity of Trigonometric functions

Periodicity of Trigonometric functions

A function f is periodic if there is a constant $p>0$ such that $f(x+p)=f(x)$ for every x in its domain — so that $f(x)=f(x+p)=f(x+2p)=\dots=f(x-p)=f(x-2p)=\dots$. The smallest such positive p is called the fundamental period (or simply the period) of f.

Rotating an angle by one full revolution ($2\pi$ radians) brings the terminal ray — and hence the point P and every coordinate-based trigonometric value — back to exactly where it started, so $\sin(x+2\pi)=\sin x$ for every x. Since no smaller positive p works (sine strictly increases then decreases within each $2\pi$ stretch), sinx has period $2\pi$; the same reasoning gives cosx, cosecx and secx period $2\pi$.

Tangent and cotangent are different: rotating by half a revolution ($\pi$) sends P to the diametrically opposite point $(-x,-y)$, and since $\tan\theta=y/x$, both signs flip and cancel: $\tan(x+\pi) = \dfrac{-y}{-x} = \dfrac{y}{x} = \tan x$ for all x. So tanx and cotx have the shorter period $\pi$.

Summary table

Solved Example 1 — find sin(41π/4)

Step 1. Write $\tfrac{41\pi}{4}$ as a whole multiple of the period $2\pi=\tfrac{8\pi}{4}$ plus a remainder: $\tfrac{41\pi}{4} = \tfrac{40\pi}{4}+\tfrac{\pi}{4} = 10\pi+\tfrac{\pi}{4}$, and $10\pi = 5\times2\pi$ is five full periods. …