Symmetry and periodicity of functions

A periodic function

A complete revolution around the unit circle (a distance of 2π) brings us back to exactly the same point. From this, we can deduce that, for any real number x and any integer k:

cos(x + 2π) = cos(x) and sin(x + 2π) = sin(x) cos(x + 2kπ) = cos(x) and sin(x + 2kπ) = sin(x)

We say that sine and cosine are periodic functions, with period 2π. A practical consequence of this is that it suffices to study these functions over an interval of length 2π, for example [−π; π], to understand their behaviour throughout the entire real line.

An even function, an odd function

The image of -x is the point symmetric to the image of x with respect to the x-axis. This gives, for any real number x:

cos(-x) = cos(x) -> cosine is EVEN sin(-x) = -sin(x) -> sine is ODD

Graphical consequences

Function	Parity	Symmetry of the graph
cosine	even	about the y-axis
sine	odd	about the origin of the coordinate system

Example

cos(-pi/3) = cos(pi/3) = 1/2 sin(-pi/3) = -sin(pi/3) = -sqrt(3)/2

Common pitfall

Do not confuse the two concepts: periodicity relates to adding 2π to x, whilst parity relates to the change in the sign of x. A common mistake is to write sin(-x) = sin(x): this is incorrect; the minus sign in front of sin(x) is missing.