91Ó°ÊÓ

The spherical coordinates are defined in terms of $\mathit{r}\mathbf{,}\mathit{θ}\mathbf{,}\mathit{ϕ}$, where r is the distance from origin, $\mathit{θ}$is the polar angle and $\mathit{ϕ}$is the azimuthal angle.

The spherical coordinates is drawn as,

In the triangle OMP, the angle M is $90°$and the length of OP is r . From the figure, it can be written as $OM=r\sin \mathrm{θ}$. Also from the figure, we can write as,

$$\begin{gathered} x=\left(r\sin \mathrm{θ}\right)\cos \mathrm{ϕ} \\ y=\left(r\sin \mathrm{θ}\right)\sin \mathrm{ϕ} \\ r=r\cos \mathrm{θ} \end{gathered}$$

Substitute localid="1657524559006" $\left(r\sin \mathrm{θ}\right)\cos \mathrm{ϕ}$for $x$, $\left(r\sin \mathrm{θ}\right)\sin \mathrm{ϕ}$for $y$and $r\cos \mathrm{θ}$for into $x^2+y^2+z^2$.

$$\begin{gathered} x^2+y^2+z^2={\left[(r\sin \mathrm{θ})\cos \mathrm{θ}\right]}^2+{\left[(r\sin \mathrm{θ})\sin \mathrm{θ}\right]}^2+{\left[(r\cos \mathrm{θ})\right]}^2 \\ =r^3\left[{\sin }^2\mathrm{θ}({\cos }^2\mathrm{ϕ}+{\sin }^2\mathrm{θ})+\left({\cos }^2\mathrm{ϕ}\right)\right] \\ =r^2\left[{\sin }^2\mathrm{θ}\left(1\right)+{\cos }^2\mathrm{ϕ}\right] \\ =r^2 \end{gathered}$$

Thus, the formula of $r$is obtained to be equal to$r=\sqrt{x^2+y^2+z^2}$.

The formula for z is$z=r\cos \mathrm{θ}$, which can be rearranged as $\mathrm{θ}={\cos }^{-1}\left(\frac{z}{r}\right)$.

Substitute $r=\sqrt{x^2+y^2+z^2}$ for $r$ into$\mathrm{θ}={\cos }^{-1}\left(\frac{z}{r}\right)$.

localid="1657529438426" $\mathrm{θ}={\cos }^{-1}\left(\frac{z}{\sqrt{x^2+y^2+z^2}}\right)$

Thus formula for $\mathrm{θ}$is obtained as $\mathrm{θ}={\cos }^{-1}\left(\frac{z}{\sqrt{x^2+y^2+z^2}}\right)$.

Now, Divide the formula of$y=(r\sin \mathrm{θ})\sin \mathrm{ϕ}\mathrm{by}x=(r\sin \mathrm{θ})\cos \mathrm{ϕ}\mathrm{as},$

$$\begin{gathered} \frac{y}{x}=\frac{(r\sin \mathrm{θ})\sin \mathrm{ϕ}}{\mathbf{(}rsin\mathrm{θ}\mathbf{)}cos\mathrm{ϕ}} \\ =\frac{\sin \mathrm{ϕ}}{\cos \mathrm{ϕ}} \\ =\tan \mathrm{ϕ} \end{gathered}$$

Therefore, $\frac{y}{x}=\tan \mathrm{Ï•}$ , then value of $\mathrm{Ï•}$can be obtained as,

$$\begin{gathered} \frac{y}{x}=\tan \mathrm{Ï•} \\ \mathrm{Ï•}={\tan }^{-1}\left(\frac{y}{x}\right) \end{gathered}$$

Therefore, the value of $\mathrm{Ï•}$is obtained as $\mathrm{Ï•}={\tan }^{-1}\left(\frac{y}{x}\right)$.