Q 47. - Calculus

Question

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.

$r={\sin }^3\theta$

Step-by-Step Solution

Answer

The required equation is ${\left(x^2+y^2\right)}^2=y^3$.

Step 1. Given information.

The given equation in polar coordinates is:

$r={\sin }^3\theta$

Step 2. Find the equation in rectangular coordinates.

$$\begin{gathered} r={\sin }^3\theta \\ r={\left(\frac{y}{r}\right)}^3 \left[\frac{y}{r}=\sin \theta \right] \\ r={\frac{y}{r^3}}^3 \end{gathered}$$

Now cross multiply,

$$\begin{gathered} r^4=y^3 \\ {\left(\sqrt{x^2+y^2}\right)}^4=y^3 \left[r^2=x^2+y^2 \mathrm{and} r=\sqrt{x^2+y^2}\right] \\ {\left(x^2+y^2\right)}^{\frac{4}{2}}=y^3 \\ {\left(x^2+y^2\right)}^2=y^3 \end{gathered}$$

Therefore, the equation in rectangular coordinates is ${\left(x^2+y^2\right)}^2=y^3$.