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### Section 12.13 : Spherical Coordinates

7. Identify the surface generated by the given equation : $$\displaystyle \varphi = \frac{{4\pi }}{5}$$

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Okay, as we discussed this type of equation in the notes for this section. We know that all points on the surface generated must be of the form $$\left( {\rho ,\theta ,\frac{{4\pi }}{5}} \right)$$.

So, we can rotate around the $$z$$-axis as much as want them to (i.e. $$\theta$$ can be anything) and we can move as far as we want from the origin (i.e. $$\rho$$ can be anything). All we need to do is make sure that the point will always make an angle of $$\frac{{4\pi }}{5}$$ with the positive ­z-axis.

In other words, we have a cone. It will open downwards and the “wall” of the cone will form an angle of $$\frac{{4\pi }}{5}$$ with the positive $$z$$-axis and it will form an angle of $$\frac{\pi }{5}$$ with the negative $$z$$-axis.