Chapter 9.6, Problem 68E

(a)

To determine

To show: the equation of the rotated graph is $r=f\left(-\cos \theta \right)$ .

Explanation of Solution

Given:

  $r=f\left(\sin \theta \right)$

Graph is rotated counterclockwise $\pi /2$ radians

Proof:

Given: $r=f\left(\cos \theta \right)$

Replace $\theta$ with $\theta -\frac{\pi }{2}$

The function will be:

  $r=f\left(\sin \left[\theta -\frac{\pi }{2}\right]\right)$

Using difference of angles formula,

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

Therefore,

  $\begin{array}{l}\sin \left(\theta -\frac{\pi }{2}\right)=\sin \theta \cos \frac{\pi }{2}-\sin \frac{\pi }{2}\cos \theta \\ \sin \left(\theta -\frac{\pi }{2}\right)=\sin \theta \left(0\right)-\left(1\right)\cos \theta \\ \sin \left(\theta -\frac{\pi }{2}\right)=-\cos \theta \end{array}$

The equation will be: $r=f\left(-\cos \theta \right)$

Conclusion:

The equation of the rotated graph is $r=f\left(-\cos \theta \right)$ .

(b)

To determine

To show: the equation of the rotated graph is $r=f\left(-\sin \theta \right)$ .

Explanation of Solution

Given:

Graph is rotated counterclockwise $\pi$ radians

Proof:

Given: $r=f\left(\sin \theta \right)$

Replace $\theta$ with $\theta -\pi$

The function will be:

  $r=f\left(\sin \left[\theta -\pi \right]\right)$

Using difference of angles formula,

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

Therefore,

  $\begin{array}{l}\sin \left(\theta -\pi \right)=\sin \theta \cos \pi -\sin \pi \cos \theta \\ \sin \left(\theta -\pi \right)=\sin \theta \left(-1\right)-\left(0\right)\cos \theta \\ \sin \left(\theta -\pi \right)=-\sin \theta \end{array}$

The equation will be:

  $r=f\left(-\sin \theta \right)$

Conclusion:

Therefore, the equation of the rotated graph is $r=f\left(-\sin \theta \right)$ .

(c)

To determine

To show: the equation of the rotated graph is $r=f\left(\cos \theta \right)$ .

Answer to Problem 68E

Explanation of Solution

Given:

Graph is rotated counterclockwise $3\pi /2$ radians

Proof:

Given: $r=f\left(\sin \theta \right)$

Replace $\theta$ with $\theta -\frac{3\pi }{2}$

The function will be:

  $r=f\left(\sin \left[\theta -\frac{3\pi }{2}\right]\right)$

Using difference of angles formula,

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

Therefore,

  $\begin{array}{l}\sin \left(\theta -\frac{3\pi }{2}\right)=\sin \theta \cos \frac{3\pi }{2}-\sin \frac{3\pi }{2}\cos \theta \\ \sin \left(\theta -\frac{3\pi }{2}\right)=\sin \theta \left(0\right)-\left(-1\right)\cos \theta \\ \sin \left(\theta -\frac{3\pi }{2}\right)=\cos \theta \end{array}$

The equation will be:

  $r=f\left(\cos \theta \right)$

Conclusion:

Therefore, the equation of the rotated graph is $r=f\left(\cos \theta \right)$ .