Simple Harmonic Motion Question [closed]

Question

I have two formulae:

Displacement = amplitude * sin(angular velocity * time)

Velocity = amplitude * angular velocity * cos(angular velocity * time)

Equivalently,

$x = A\sin(\omega t)$

$v = A\omega\cos(\omega t)$

And the question is using these two formulae show $v = \pm \omega\sqrt{A^2 - x^2}$

Answer 1

Calculate $x^2 +\frac{v^2}{\omega ^2}$:

$x^2 + \frac{v^2}{\omega ^2}=A^2(\cos^2(\omega t) + \sin^2(\omega t) )$

$x^2+ \frac{v^2}{\omega ^2}=A^2$

$v^2=\omega ^2(A^2-x^2)$

Which gives us

$v=\pm \omega \sqrt{A^2-x^2}$