Chapter 8.2, Problem 31E

To determine

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

  $r=-\cos 5\theta$

Answer to Problem 31E

The graph of polar equation is

  

The equation in rectangular coordinates is .

Explanation of Solution

Given: Polar equation: $r=-\cos 5\theta$

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

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

  $\cos 3\theta =4{\cos }^3\theta -3\cos \theta$

  $\cos 2\theta =2{\cos }^2\theta -1$

  $\sin 3\theta =3\sin \theta -4{\sin }^3\theta$

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

Calculation:

Given Polar equation is $r=-\cos 5\theta$

Drawing the polar-form curve for $r=-\cos 5\theta$ on a graph as per the above definition of polar equation, we have:

  

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

  $r=-\cos 5\theta$

  $\begin{array}{l}r=-\cos \left(2\theta +3\theta \right)\\ r=-\cos 2\theta \cos 3\theta +\sin 2\theta sin3\theta \end{array}$

  $r=\left(2{\cos }^2\theta -1\right)\left(4{\cos }^3\theta -3\cos \theta \right)-2\sin \theta \cos \theta \left(3\sin \theta -4{\sin }^3\theta \right)$

From the above given formulas, we have:

Conclusion:

Hence, the graph of polar equation is

  

The equation in rectangular coordinates is .