Is $y(x)=\alpha \sin^2(\beta x+\gamma)$ a sinusoidal function?

Question

Is $$y(x)=\alpha \sin^2(\beta x+\gamma)$$ a sinusoidal function? I know it can be alternatively represented as $$y(x)=\frac12 (\alpha-\alpha \cos(2\beta x+2 \gamma))$$ Can this be transformed to $$y(x)=A \sin(b x+c)$$? My intuition says no, but my physics teacher named it a sinudoidal function.

Answer 1

No. To answer the second question, your function is $0$ at points whose separation is $\pi/2$ (namely $x = 0, \pm \pi/2, \pm \pi, \ldots$) while the thing you want to transform it to has zeros separated by $\pi$.

To answer the larger question, "yes". "Sinusoidal" usually means "shaped like a sine function, that's been translated and scaled in either (or both of) x and y." So your teacher's statement was correct, but perhaps s/he should have more precisely defined "sinusoidal".

(My answer no longer makes sense, because the problem's been edited; in the original, it was just $y = \sin^2(x)$. The same general point remains.)