Is the phase shift of $-3\sin\left(2x+\frac{\pi}{2}\right)+1$ equal to $-\frac{\pi}{4}$ or just $-\frac{\pi}{2}$?

Question

It's kind of a silly question, but ...

For the function $$-3\sin\left(2x+\frac{\pi}{2}\right)+1$$ is its phase shift $-\dfrac{\pi}{4}$ or just $-\dfrac{\pi}{2}$?

Answer 1

The definition of "phase shift" actually depends on the context. For example, in a context where $A\sin(\omega x - \phi)$ is a sine wave or signal and $x$ is time, one often calls $\phi$ or $-\phi$ the phase shift. To be more precise, this could be called an angular phase shift, since this parameter shifts the argument of the sine function (an angle) away from its original angle.

On the other hand, in other contexts one would rather write the same sine wave as $A\sin\left(\omega(x - \frac{\phi}{\omega})\right)$, which highlights a different kind of shift, namely a temporal shift of $\frac{\phi}{\omega}$ seconds. If you graph the sine wave as a function of time, then this quantity is the proper horizontal shift that you see on the graph.

Answer 2

The phase shift of $-3\sin(2x+\frac{\pi}{2})+1=-3\sin\left(2\left(x+\frac{\pi}{4}\right)\right)+1$ relative to $-3\sin(2x)+1$ is $-\frac{\pi}{4}$.