Using small angle approximation to turn x(t) and y(t) to the usual projectile kinematic equation

Question

Given the horizontal and vertical position-time functions:
\begin{align*}
x(t) &= \frac{v_0\cos\ \theta\sin\ \omega t}{\omega}\\
y(t) &= \dfrac{v_0\sin\ \theta\sin(\omega t)}{\omega} + \frac{mg}{k} \cos(\omega t) – \frac{mg}{k}
\end{align*}

Show that both functions reduce to usual projectile kinematic equations for very small $\omega$.

Answer 1

Use $$\sin x\approx x\\\cos x\approx 1-\frac12x^2$$ So $$\frac{\sin(\omega t)}{\omega}\approx t$$and $$\cos(\omega t)-1\approx-\frac12(\omega t)^2$$