Finding the values of $n$ for which $\sin(πn/2) = 1$

Question

I am trying to do a proof and I want to find the value(s) of $n$ for which the function $f(n) = \sin(πn/2)$ is at its maximum value. I know that the maximum value which $\sin(πn/2)$ can be is $1$, but I am not sure how to find the value(s) of $n$ for which $f(n)$ is equal to $1$. Any insights are appreciated.

Answer 1

Hints:

• $\sin(x)$ is a periodic function, with period $2\pi$. Thus, $\sin(x) = \sin(x + 2\pi) = \sin(x + 4\pi) = ... = \sin(x + 2k\pi)$ for all integers $k$.

• We know that the maximum of $\sin(x)$ is attained at $x = \pi/2$ (where $\sin(x) = 1$).

Combining these two facts, what does this suggest $n$ could be?

Answer 2

Hint:

$$\dfrac{n\pi}2=2m\pi+\dfrac\pi2$$ where $m$ is any integer

$\implies n=?$