Chapter 8.2, Problem 29E

To determine

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

  $r=\sin 2\theta$

Answer to Problem 29E

The graph of polar equation is

  

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

Explanation of Solution

Given: Polar equation: $r=\sin 2\theta$

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 $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

  $\sin 2\theta =2\sin \theta \cos \theta$

Calculation:

Given Polar equation is $r=\sin 2\theta$

Drawing the polar-form curve for $r=\sin 2\theta$ on a graph as per the above definition of polar equation, we have:

  

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

  $r=\sin 2\theta$

  $r=2\sin \theta \cos \theta$

Multiplying both the sides by $r^2$ , we have:

  $r^2\times r=\left(2\sin \theta \cos \theta \right)\times r^2$

  $r^3=2\left(r\sin \theta \right)\left(r\cos \theta \right)$

From the above given formulas, we have:

  ${\left(x^2+y^2\right)}^{3/2}=2xy$

Conclusion:

Hence, the graph of polar equation is

  

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