Chapter 9

Graph the family of polar equations \(r=1+\sin n \theta\) for \(n=1,2,3,4,\) and \(5 .\) How is the number of loops related to \(n ?\)

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ 4(\sqrt{3}+i) $$

43- 48 . Use a graphing device to draw the curve represented by the parametric equations. $$ x=\sin (\cos t), \quad y=\cos \left(t^{3 / 2}\right), \quad 0 \leq t \leq 2 \pi $$

Convert the equation to polar form. $$ x^{2}-y^{2}=1 $$

Graph the family of polar equations \(r=1+c \sin 2 \theta\) for \(c=0.3,0.6,1,1.5,\) and \(2 .\) How does the graph change as \(c\) increases?

43- 48 . Use a graphing device to draw the curve represented by the parametric equations. $$ x=\sin (\cos t), \quad y=\cos \left(t^{3 / 2}\right), \quad 0 \leq t \leq 2 \pi $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ -3-3 i $$

Convert the polar equation to rectangular coordinates. $$ r=7 $$

\(49-52\) . A polar equation is given. (a) Express the polar equation in parametric form. (b) Use a graphing device to graph the parametric equations you found in part (a). $$ r=2^{\theta / 12}, \quad 0 \leq \theta \leq 4 \pi $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ 2+i $$

Convert the polar equation to rectangular coordinates. $$ r=-3 $$

\(49-52\) . A polar equation is given. (a) Express the polar equation in parametric form. (b) Use a graphing device to graph the parametric equations you found in part (a). $$ r=\sin \theta+2 \cos \theta $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ 3+\sqrt{3} i $$

\(49-52\) . A polar equation is given. (a) Express the polar equation in parametric form. (b) Use a graphing device to graph the parametric equations you found in part (a). $$ r=\frac{4}{2-\cos \theta} $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ \sqrt{2}+\sqrt{2} i $$

Convert the polar equation to rectangular coordinates. $$ \theta=\pi $$

\(49-52\) . A polar equation is given. (a) Express the polar equation in parametric form. (b) Use a graphing device to graph the parametric equations you found in part (a). $$ r=2^{\sin \theta} $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ -\pi i $$

Convert the polar equation to rectangular coordinates. $$ r \cos \theta=6 $$

Sketch a graph of the rectangular equation. [Hint: First convert the equation to polar coordinates.] $$ \left(x^{2}+y^{2}\right)^{3}=4 x^{2} y^{2} $$

Find the product \(z_{1} z_{2}\) and the quotient \(z_{1} / z_{2}\) . Express your answer in polar form. $$ z_{1}=\cos \pi+i \sin \pi, \quad z_{2}=\cos \frac{\pi}{3}+i \sin \frac{\pi}{3} $$

Convert the polar equation to rectangular coordinates. $$ r=2 \csc \theta $$

Sketch a graph of the rectangular equation. [Hint: First convert the equation to polar coordinates.] $$ \left(x^{2}+y^{2}\right)^{3}=\left(x^{2}-y^{2}\right)^{2} $$

Find the product \(z_{1} z_{2}\) and the quotient \(z_{1} / z_{2}\) . Express your answer in polar form. $$ z_{1}=\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}, \quad z_{2}=\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4} $$

Convert the polar equation to rectangular coordinates. $$ r=4 \sin \theta $$

Sketch a graph of the rectangular equation. [Hint: First convert the equation to polar coordinates.] $$ \left(x^{2}+y^{2}\right)^{2}=x^{2}-y^{2} $$

Find the product \(z_{1} z_{2}\) and the quotient \(z_{1} / z_{2}\) . Express your answer in polar form. $$ z_{1}=3\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right), \quad z_{2}=5\left(\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}\right) $$

Convert the polar equation to rectangular coordinates. $$ r=6 \cos \theta $$

Sketch a graph of the rectangular equation. [Hint: First convert the equation to polar coordinates.] $$ x^{2}+y^{2}=\left(x^{2}+y^{2}-x\right)^{2} $$

Find the product \(z_{1} z_{2}\) and the quotient \(z_{1} / z_{2}\) . Express your answer in polar form. $$ z_{1}=7\left(\cos \frac{9 \pi}{8}+i \sin \frac{9 \pi}{8}\right), \quad z_{2}=2\left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right) $$

Convert the polar equation to rectangular coordinates. $$ r=1+\cos \theta $$

Show that the graph of \(r=a \cos \theta+b \sin \theta\) is a circle, and find its center and radius.

Find the product \(z_{1} z_{2}\) and the quotient \(z_{1} / z_{2}\) . Express your answer in polar form. $$ \begin{array}{l}{z_{1}=4\left(\cos 120^{\circ}+i \sin 120^{\circ}\right)} \\\ {z_{2}=2\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)}\end{array} $$

Convert the polar equation to rectangular coordinates. $$ r=3(1-\sin \theta) $$

(a) Graph the polar equation \(r=\tan \theta \sec \theta\) in the viewing rectangle \([-3,3]\) by \([-1,9]\) . (b) Note that your graph in part (a) looks like a parabola (see Section \(3.5 ) .\) Confirm this by converting the equation to rectangular coordinates.

Find the product \(z_{1} z_{2}\) and the quotient \(z_{1} / z_{2}\) . Express your answer in polar form. $$ \begin{array}{l}{z_{1}=\sqrt{2}\left(\cos 75^{\circ}+i \sin 75^{\circ}\right)} \\\ {z_{2}=3 \sqrt{2}\left(\cos 60^{\circ}+i \sin 60^{\circ}\right)}\end{array} $$

Convert the polar equation to rectangular coordinates. $$ r=1+2 \sin \theta $$

Orbit of a Satellite Scientists and engineers often use polar equations to model the motion of satellites in earth orbit. Let's consider a satellite whose orbit is modeled by the equation \(r=22500 /(4-\cos \theta),\) where \(r\) is the distance in miles between the satellite and the center of the earth and \(\theta\) is the angle shown in the following figure. (a) On the same viewing screen, graph the circle \(r=3960\) (to represent the earth, which we will assume to be a sphere of radius 3960 \(\mathrm{mi}\) and the polar equation of the satellite's orbit. Describe the motion of the satellite as \(\theta\) increases from 0 to \(2 \pi .\) (b) For what angle \(\theta\) is the satellite closest to the earth? Find the height of the satellite above the earth's surface for this value of \(\theta .\)

Find the product \(z_{1} z_{2}\) and the quotient \(z_{1} / z_{2}\) . Express your answer in polar form. $$ \begin{array}{l}{z_{1}=4\left(\cos 200^{\circ}+i \sin 200^{\circ}\right)} \\\ {z_{2}=25\left(\cos 150^{\circ}+i \sin 150^{\circ}\right)}\end{array} $$

Convert the polar equation to rectangular coordinates. $$ r=2-\cos \theta $$

Find the product \(z_{1} z_{2}\) and the quotient \(z_{1} / z_{2}\) . Express your answer in polar form. $$ \begin{array}{l}{z_{1}=\frac{4}{5}\left(\cos 25^{\circ}+i \sin 25^{\circ}\right)} \\ {z_{2}=\frac{1}{5}\left(\cos 155^{\circ}+i \sin 155^{\circ}\right)}\end{array} $$

Convert the polar equation to rectangular coordinates. $$ r=\frac{1}{\sin \theta-\cos \theta} $$

A Transformation of Polar Graphs How are the graphs of $$ r=1+\sin \left(\theta-\frac{\pi}{6}\right) $$ and \(\quad r=1+\sin \left(\theta-\frac{\pi}{3}\right)\) related to the graph of \(r=1+\sin \theta ?\) In general, how is the graph of \(r=f(\theta-\alpha)\) related to the graph of \(r=f(\theta) ?\)

Write \(z_{1}\) and \(z_{2}\) in polar form, and then find the product \(z_{1} z_{2}\) and the quotients \(z_{1} / z_{2}\) and 1\(/ z_{1}\) . $$ z_{1}=\sqrt{3}+i, \quad z_{2}=1+\sqrt{3} i $$

Convert the polar equation to rectangular coordinates. $$ r=\frac{1}{1+\sin \theta} $$

Choosing a Convenient Coordinate System Compare the polar equation of the circle \(r=2\) with its equation in rectangular coordinates. In which coordinate system is the equation simpler? Do the same for the equation of the four- leaved rose \(r=\sin 2 \theta .\) Which coordinate system would you choose to study these curves?

Write \(z_{1}\) and \(z_{2}\) in polar form, and then find the product \(z_{1} z_{2}\) and the quotients \(z_{1} / z_{2}\) and 1\(/ z_{1}\) . $$ z_{1}=\sqrt{2}-\sqrt{2} i, \quad z_{2}=1-i $$

Convert the polar equation to rectangular coordinates. $$ r=\frac{4}{1+2 \sin \theta} $$

Choosing a Convenient Coordinate System Compare the rectangular equation of the line \(y=2\) with its polar equation. In which coordinate system is the equation simpler? Which coordinate system would you choose to study lines?

Write \(z_{1}\) and \(z_{2}\) in polar form, and then find the product \(z_{1} z_{2}\) and the quotients \(z_{1} / z_{2}\) and 1\(/ z_{1}\) . $$ z_{1}=2 \sqrt{3}-2 i, \quad z_{2}=-1+i $$