I've got the following problem which is solved in this picture: https://image.prntscr.com/image/atu-SdJ9QM_c7LShqdYf1g.png
Can somebody explain to me why the first phase we get from the first equation is not correct (phase = -pi/4), and we must calculate the other one (phase = -3pi/4)?
When solving for the phase shift, the solution uses algebra and more specifically the $arcsin(x)$ function, which can have multiple outputs for the same input. From there, the author looks at the graph and realizes that the picture of the function shows that it hits the bottom most point in its period right after it starts from zero and goes rightward.
Now on the unit circle, when going in the traditional counter-clockwise direction, from which value, $\frac{-3\pi}{4}$ or $\frac{-\pi}{4}$, do you first hit the minimum value for sine and then proceed? Obviously $\frac{-3\pi}{4}$.
The first answer is not right because it does not satisfy all parts of the solution. We want it to have a certain $arcsin(x)$ value at zero, but we also want it to be right before the minimum, and the latter criterion is only considered later.