(1) Find the (smallest) period of and .
(2) Show that these functions also have period .
(3) Show that the constant is also a periodic function with period .
We know that the period of a normal or is . When there are values or variables being multiplied by this
variable, the period becomes divided by the values or variables. We know that:
Where is the period.
Using this relation, along with our variable , we can solve for the period of and as follows:
Since the period will be smallest at , plugging into equation (6.1.2) shows that the smallest period of and is:
We are also given that:
Using this relation, we can solve for the period again as follows:
We know that the period is smallest at , and plugging this value into (6.1.5) proves that:
We know that, starting at 0:
Where the period is represented by 0 to 2L. We are also given that:
Rearranging (6.1.8) allows us to solve for L:
Multiplying (6.1.9) by 2 allows us to find the period of , as follows:
This shows that is indeed a periodic function with a period of . This also shows that at any given value or period throughout the periodic function, holds its constant value.