Chapter 8, Problem 4RCC

(a)

To determine

To write: the type of curve has a polar equation of the given form.

Answer to Problem 4RCC

The type of curve for polar equation $r=a\cos \theta$ or $r=a\sin \theta$ is circle.

Explanation of Solution

Given:

  $r=a\cos \theta \text{ or }r=a\sin \theta$

Calculation:

find the type of curve which has a polar equation of the form $r=a\cos \theta$ or

  $r=a\sin \theta$ .

Since, the equation $r=a\sin \theta$ remain unchanged when $\theta$ is replace by $\pi -\theta$ , therefore its graph is symmetrical about the vertical line $\theta =\frac{\pi }{2}$ and it takes $2\pi$ radians to complete a revolution.

Similarly, $r=a\cos \theta$ is symmetrical about the polar axis, because equation remains unchanged when $\theta$ replace by is $-\theta$ and it takes $2\pi$ radians to complete a revolution.

Thus the type of curve for polar equation $r=a\cos \theta$ or $r=a\sin \theta$ is circle.

Conclusion:

Therefore, the type of curve for polar equation $r=a\cos \theta$ or $r=a\sin \theta$ is circle.

(b)

To determine

To write: the type of curve has a polar equation of the given form.

Answer to Problem 4RCC

Since, $a=b$ polar equations of the form $r=a\left(1\pm \cos \theta \right)\text{ or }r=a\left(1\pm \sin \theta \right)$ have curves of shape Cardioids.

Explanation of Solution

Given:

  $r=a\left(1\pm \cos \theta \right)\text{ or }r=a\left(1\pm \sin \theta \right)$

Calculation:

Find the type of curve which has a polar equation of the form $r=a\left(1\pm \cos \theta \right)$ or $r=a\left(1\pm \sin \theta \right)$ .

For $r=a\pm b\cos \theta \text{ or }r=a\pm b\sin \theta$ , curve is Limacons.

Since, $a=b$ polar equations of the form $r=a\left(1\pm \cos \theta \right)\text{ or }r=a\left(1\pm \sin \theta \right)$ have curves of shape Cardioids.

Conclusion:

Since, $a=b$ polar equations of the form $r=a\left(1\pm \cos \theta \right)\text{ or }r=a\left(1\pm \sin \theta \right)$ have curves of shape Cardioids.

(c)

To determine

To write: the type of curve has a polar equation of the given form.

Answer to Problem 4RCC

The above polar equations have different types of Limacons.

Explanation of Solution

Given:

  $r=a\pm b\cos \theta \text{ or }r=a\pm b\sin \theta$

Calculation:

Find the type of curve which has a polar equation of the form $r=a\pm b\cos \theta$ or $r=a\pm b\sin \theta$ .

As there is no any relation between $\text{'}a\text{'}$ and $\text{'}b\text{'}$ or their sign. The orientation depends on the trigonometric functions (sine or cosine) and the sign of b . Therefore the above polar equations have different types of Limacons.

Conclusion:

Therefore the above polar equations have different types of Limacons.

(d)

To determine

To write: the type of curve has a polar equation of the given form.

Answer to Problem 4RCC

The curve of these types of polar equations is of from of Rose.

Explanation of Solution

Given:

  $r=a\cos n\theta \text{ or }r=a\sin n\theta$

Calculation:

Find the type of curve which has a polar equation of the form $r=a\cos n\theta$ or

  $r=a\sin n\theta$ .

The curve of these types of polar equations is of from of Rose.

There are n -leaves in Rose, if n is odd or $2n$ leaves, if n is even.

Conclusion:

Therefore, the curve of these types of polar equations is of from of Rose.