Change in Angle Between Vector and Positive x axis

Question

I have a vector in the $xy$ plane, $\vec{A}$, defined by $\|\vec{A}\|=8$, and $\theta=130^\circ$ with the positive $x$ axis. This gives the component form $\langle 8\cos130^\circ, 8\sin130^\circ, 0\rangle$. Now I need to find the spherical coordinates for $\vec{A}+3\hat{k}$, which are $\langle\|\vec{A}+3\hat{k}\|, \theta,\phi\rangle$. In order to evaluate $\theta$, I used the formula $$\theta=\cos^{-1}\left(\frac{A_x}{\|\vec{A}+3\hat{k}\|}\right)=\cos^{-1}\left(\frac{8\cos130^\circ}{\sqrt{(8\cos130^\circ)^2+(8\sin130^\circ)^2+3^2}}\right)\approx 127^\circ.$$ Why is it that I do not get the original $\theta=130^\circ$? Does the angle made with the positive $x$ axis change with the addition of a $z$ component?

Answer 1

If $\theta$ in your spherical system is going to have the same meaning as $\theta$ in the polar coordinates, then it should be $130$. The change formula is $$ \theta_{\rm spherical}=\arctan\left(\frac yx\right)=\arctan\left(\frac{8\sin 130}{8\cos 130}\right)=\arctan(\tan(130))=130. $$