Question

how many possible values for y are there where y=arctan 3?

Answer 1

\[\bf y=\arctan(3) \implies \tan(y)=3\]Note that since \(\bf y\) is initially defined for the inverse tangent function then \(\bf -\pi/2 < y < \pi/2\).

Now if we graph \(\bf tan(y)\) for \(\bf -\pi/2 < y < \pi/2\), we get a single cycle of tangent and we will only find a single value for which the graph will go through 3.

Hence there is only 1 possible value for \(\bf y\) such that \(\bf y=arctan(3)\).