Chapter 8.2, Problem 32E

To determine

To sketch the graph of the given polar equation and express the equation in rectangular coordinates −

  $r=\sin 4\theta$

Answer to Problem 32E

The graph of polar equation is

  

The equation in rectangular coordinates is ${\left(x^2+y^2\right)}^{5/2}=4x^{3}y-4x{y}^3$ .

Explanation of Solution

Given: Polar equation: $r=\sin 4\theta$

Formula Used:

A polar equation is any equation that describes a relation between r and θ , where r represents the distance from the pole (origin) to a point on a curve, and θ represents the counter-clockwise angle made by a point on a curve, the pole, and the positive x-axis.

Also, $x=r\cos \theta$ and $y=r\sin \theta$

  $r=\sqrt{x^2+y^2}$ and $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

  $\sin \left(A+B\right)=\sin A\cos B+\cos A\sin B$

  $\sin 2A=2\sin A\cos A$

  $\cos 2A={\cos }^{2}A-{\sin }^{2}A$

Calculation:

Given Polar equation is $r=\sin 4\theta$

Drawing the polar-form curve for $r=\sin 4\theta$ on a graph as per the above definition of polar equation, we have:

  

Now, to express the equation in rectangular coordinates, we have:

  $r=\sin 4\theta$

  $\begin{array}{l}r=\sin 2\theta \cos 2\theta +\cos 2\theta \sin 2\theta \\ r=2\sin 2\theta \cos 2\theta \\ r=4\sin \theta \cos \theta \times \left({\cos }^2\theta -{\sin }^2\theta \right)\\ r=4\sin \theta {\cos }^3\theta -4{\sin }^3\theta \cos \theta \end{array}$

Multiplying both the sides by $r^4$ , we have:

  $\begin{array}{l}{r}^4\times r=\left(4\sin \theta {\cos }^3\theta -4{\sin }^3\theta \cos \theta \right)\times r^4\\ r^5=4\left(r\sin \theta \right){\left(r\cos \theta \right)}^3-4{\left(r\sin \theta \right)}^3\left(r\cos \theta \right)\end{array}$

From the above given formulas, we have:

  $\begin{array}{l}{\left(x^2+y^2\right)}^{5/2}=4y{x}^3-4y^{3}x\\ {\left(x^2+y^2\right)}^{5/2}=4x^{3}y-4x{y}^3\end{array}$

Conclusion:

Hence, the graph of polar equation is

  

The equation in rectangular coordinates is ${\left(x^2+y^2\right)}^{5/2}=4x^{3}y-4x{y}^3$ .