Q:

max and min sine function and all intervals

I have a calculus question:
The voltage signal from a standard North
American wall socket can be described by the
equation V(t) = 170sin(120πt), where t is time,
in seconds, and V(t) is the voltage, in volts, at
time t.
a) Find the maximum and minimum voltage
levels, and the times at which they occur.
Basically I understand that my maximum is 170 Volts, my minimum is -170 Volts but it is asking for the exact times at which they occur. So my answer turns out to be :
t = {t | t = (4k+1)/240, k >= 0, k ∈ Z}  <- Max
t = {t | t = (4k+1)/240, k >= 0, k ∈ Z}  <- Min
I am VERY confused as to how those above intervals came to be and if for example the question was a cosine function instead of sine, would it be different?
Thank you very much everyone!

A:

You are correct to recognize that the amplitude is 170 so the maximum will be 170 and the minimum will be -170.
$\sin(t)$ has a period of $2\pi$ and a maximum at every $\frac{\pi}{2}+2\pi *k$ and a minimum at every $\frac{3\pi}{2}+2\pi*k$ where $k$ is a constant.
The major difference between the functions is the period. In your function, $V(t)=170\sin(120\pi t)$, the coefficient of $t$ is $120\pi$, which means that the function travels $120\pi$ times as fast, or completes a period $120\pi$ times faster than usual. Since $\sin(t)$ has a period of $2\pi$, then $V(t)$ will make one period in $\frac{2\pi}{120\pi}$ or every $\frac{1}{60}$. 
Recall that $\sin(t)$ has a maximum every $\frac{1}{4}$ of the way along a period. This means that the first maximum will be reached at $\frac{1}{4}*\frac{1}{60}=\frac{1}{240}$. Since the function has a period of $\frac{1}{60}$, then you can continuously add $\frac{1}{60}$ to $\frac{1}{240}$ to obtain more maxima, which would then mean maximums are when:
$$t=\frac{4k+1}{240}$$
where $k$ is an integer greater than or equal to $0$.
Since a minimum occurs every $\frac{3}{4}$ of a period, using similar logic we find that the minima are at:
$$t=\frac{4k+3}{240}$$
again where $k$ is an integer greater than or equal to $0$.

