Find fundamental period of x(t) = cos(Pi * t)

[VinnyCee]

Homework Statement

Find fundamental period of $$x(t)\,=\,cos\left(\pi\,t\right)$$

Homework Equations

$$x(t)\,=\,A\,sin\left(\omega_0\,t\,+\,\phi\right)$$

Which has a fundamental period $$T\,=\,\frac{2\pi}{\omega_0}$$

The Attempt at a Solution

$$\omega_0\,=\,\pi$$ <---- Right?

$$T\,=\,\frac{2\pi}{\pi}\,=\,2$$

$$T\,=\,2$$

 

[dextercioby]

T=2 s. Don't forget the units.

 

[HallsofIvy]

T=2 s. Don't forget the units.

"s"? What is "s"? The problem, as stated, does not have units- it is a pure function. Even if you assume "t" is time (I would not, I see no reason to assume this is a physics problem rather than a mathematics problem) why would you assume the units are seconds rather than minutes or hours?

In any case, VinnyCee, your analysis is correct.

 

[VinnyCee]

Thanks for your help!

So, are the following correct?

$$x(t)\,=\,cos\left(3\,\pi\,t\right)\,\,\longrightarrow\,\,\omega_0\,=\,3\,\pi\,\,\longrightarrow\,\,T\,=\,\frac{2\,\pi}{3\,\pi}\,=\,\frac{2}{3}$$

$$x(t)\,=\,sin\left(4\,\pi\,t\right)\,\,\longrightarrow\,\,\omega_0\,=\,4\,\pi\,\,\longrightarrow\,\,T\,=\,\frac{2\,\pi}{4\,\pi}\,=\,\frac{1}{2}$$

$$x(t)\,=\,cos\left(\frac{\pi}{2}\,t\right)\,\,\longrightarrow\,\,\omega_0\,=\,\frac{\pi}{2}\,\,\longrightarrow\,\,T\,=\,\frac{2\,\pi}{\frac{\pi}{2}}\,=\,\frac{4\,\pi}{\pi}\,=\,4$$

$$x(t)\,=\,sin\left(\frac{\pi}{3}\,t\right)\,\,\longrightarrow\,\,\omega_0\,=\,\frac{\pi}{3}\,\,\longrightarrow\,\,T\,=\,\frac{2\,\pi}{\frac{\pi}{3}}\,=\,\frac{6\,\pi}{\pi}\,=\,6$$

 

[HallsofIvy]

Yes.