# Measuring phase constants from the sine function

In simple harmonic motion, is the phase, by definition, always measured using the sine function? I'm asking because a question came up that provided $$\omega$$ and the amplitude, and also specified the initial phase constant to be $$\pi /4$$.

Now, the answer says

$$A\sin(\omega t + \frac{\pi}{4})$$

but if you used the cosine function, it wouldn't be the same.

Physically, it does not matter whether we use $$\sin()$$ or $$\cos()$$. It's a matter of convention and you are free to choose either one.
Therefore, if somebody defined the amplitude, the frequency and the "initial phase" $$\varphi_0 = 0$$ of the oscillation, the function $$x(t)$$ is not (!) well defined. However, if we say, the initial phase is such that the harmonic oscillator starts with the maximal amplitude, we know $$x(t) = A_0 \cos(\omega t)$$. If instead we define the initial phase as the point with the minimal velocity (sign matters), we know that $$v = - A_0 \omega \cos(\omega t)$$. Hence, after integration we obtain $$x(t) = A_0 \sin(\omega t)$$.