Chapter 8.4, Problem 49E

(a)

To determine

To find: the polar equation in parametric form.

Answer to Problem 49E

The parametric form of the polar equation $r=\sin \theta +2\cos \theta$ is

  $\begin{array}{l}x=\left(\sin t+2\cos t\right)\cos t\\ y=\left(\sin t+2\cos t\right)\sin t\end{array}$

Explanation of Solution

Given:

Polar equation $r=2^{\frac{\theta }{12}},0\le \theta \le 4\pi$

Calculation:

Express the polar equation in the parametric equation.

The graph of the polar equation $r=f\left(\theta \right)$ is the same as the graph of the parametric equation

  $x=f\left(t\right)\cos t$ ,and $y=f\left(t\right)\sin t$

Since $r=2^{\frac{\theta }{12}}$ , get the parametric equation as

  $x=2^{\frac{t}{12}}\cos t$ and $y=2^{\frac{t}{12}}\sin t$ .

Conclusion:

Therefore, the parametric form of the polar equation is $r=2^{\frac{\theta }{12}}$ ,

  $x=2^{\frac{t}{12}}\cos t$ and $y=2^{\frac{t}{12}}\sin t$

(b)

To determine

To find: the graph of the parametric equations.

Answer to Problem 49E

  

Explanation of Solution

Given:

Polar equation $r=2^{\frac{\theta }{12}}$ for $0\le \theta \le 4\pi$

Calculation:

Find the graph of the parametric equations $x=2^{\frac{t}{12}}\cos t$ and $y=2^{\frac{t}{12}}\sin t$ .

To graph the given parametric equation by using a graphing calculator, enter the equation in the graphing calculator that displays the graph of the given parametric equation as shown below.

  

Conclusion:

Therefore, the graph of the parametric equations is drawn.