Periodicity

Since the point $P_\alpha$ returns to the same position after every complete turn, sine and cosine repeat at regular intervals: they are periodic functions.

Property — Periodicity

The functions $\sin$ and $\cos$ are periodic with period $2\pi$: for every $\alpha\in\mathbb{R}$ and every $k\in\mathbb{Z}$, $\sin(\alpha + 2k\pi) = \sin\alpha,\qquad \cos(\alpha + 2k\pi) = \cos\alpha.$ The tangent is instead periodic with period $\pi$: $\tan(\alpha + k\pi) = \tan\alpha.$

Proof

An increase of $2\pi$ in the angle corresponds to a complete turn on the unit circle: the point $P_\alpha$ returns to the same position, so its coordinates (sine and cosine) do not change. For the tangent it is enough to observe that $\tan(\alpha+\pi) = \dfrac{\sin(\alpha+\pi)}{\cos(\alpha+\pi)} = \dfrac{-\sin\alpha}{-\cos\alpha} = \tan\alpha$ (using the reduction formulae for associated arcs). ∎

The sign of $\sin$ depends on the ordinate, that of $\cos$ on the abscissa, that of $\tan$ on their ratio (positive in the first and third quadrants, negative in the second and fourth).

Visual table of the signs over the four quadrants.

Links

Topics: Trigonometry
Concepts: Cosine · Periodicity · Sine · Tangent
Functions: Cosine · Sine · Tangent
Skills: Proving