Q 32 - Calculus

Question

Describe the graphs of the equations  and provide alternative equations in the specified coordinate system

. Change $r=4\sin \theta$ to the rectangular and spherical systems. 

Step-by-Step Solution

Answer

The equation in rectangular system is $x^2+y^2-4y=0$ 

The equation in spherical system is $\rho =4csc\varphi \sin \theta$

Step 1:Given information

The given expression is $r=4\sin \theta$

Step 2: SImplification

To convert $r=4\sin \theta$ into rectangular coordinates systems as follows

substitute $r=\sqrt{x^2+y^2}$ and $\sin \theta =\frac{y}{\sqrt{x^2+y^2}}$ in $r=4\sin \theta$

$\Rightarrow \sqrt{x^2+y^2}=\frac{4y}{\sqrt{x^2+y^2}}$

$\Rightarrow x^2+y^2-4y=0$

Therefore the expression in rectangular system is $x^2+y^2-4y=0$

Step 3:Simplification

consider spherical coordinates as $r=\rho \sin \varphi ,\theta =\theta ,z=\rho \cos \varphi$

substituting $r=\rho \sin \varphi$ in the equation $r=4\sin \theta$

$\Rightarrow \rho \sin \varphi =4\sin \theta$

$\Rightarrow \rho =\frac{4\sin \theta }{\sin \varphi }$

$\Rightarrow \rho =4csc\varphi \sin \theta$

Therefore the equation in spherical system is $\rho =4csc\varphi \sin \theta$