what is the range of independent variable of a sinusoid?

Question

Let us say I represent a sinusoidal function as: $$ y = \sin x $$

Then what will be its time period?

How to decide what is the range of its independent variable $x$ as we don't know if it is in time (sec) or angle (degrees or radians)?

Answer 1

So to know the period you do the following.

Let $T$ be the period of $y = \sin(x).$ By definition, we have that

$$y(t) = y(t + T) \iff \sin{t} = \sin{(t+T)}.$$

Solving this equation, you have that

$$t = t + T + 2k\pi \vee t = -t - T +2k \pi, \text{ where } t \in \mathbb{Z}.$$

From this, you conclude that $T=2\pi$ (considering The least positive number.) In short, is the sine function, its period is $2\pi.$

For the second part, note that you can say that the domain in $\mathbb{R}$ even without knowing the measure you are using for the angles.

You should always specify if you are using time, degrees or radians. Because the values of your function will depend on that.

For examples, $y(\pi)$ will have different values if you consider $\pi \text{ rad},$ and $\pi \text{ degrees}.$ It’s up to you to decide what to use (unless it is stated that you shroud use a specific one), but always state it to avoid confusion.