Answer
Write $z'$ in terms of $R, y, z$.
$$z'=z\cos R+y\sin R$$
Work Step by Step
$$z'=r\sin(\theta+R)\hspace{1cm}y=r\cos\theta\hspace{1cm}z=r\sin\theta$$
We now would analyze $z'$ formula.
For $\sin(\theta+R)$, the sine sum identity can be applied:
$$\sin(\theta+R)=\sin\theta\cos R+\cos\theta\sin R$$
Therefore, $$z'=r(\sin\theta\cos R+\cos\theta\sin R)$$
$$z'=r\sin\theta\cos R+r\cos\theta\sin R$$
$$y'=(r\sin\theta)\cos R+(r\cos\theta)\sin R$$
Now recall that $z=r\sin\theta$ and $y=r\cos\theta$.
$$z'=z\cos R+y\sin R$$
That is the formula of $z'$ in terms of $R, y, z$.
