Chapter 9

Convert the polar equation to rectangular coordinates. $$ r=\frac{2}{1-\cos \theta} $$

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\(/ z_{1}\) . $$ z_{1}=-\sqrt{2} i, \quad z_{2}=-3-3 \sqrt{3} i $$

Convert the polar equation to rectangular coordinates. $$ r^{2}=\tan \theta $$

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\(/ z_{1}\) . $$ z_{1}=5+5 i, \quad z_{2}=4 $$

Convert the polar equation to rectangular coordinates. $$ r^{2}=\sin 2 \theta $$

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\(/ z_{1}\) . $$ z_{1}=4 \sqrt{3}-4 i, \quad z_{2}=8 i $$

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

Spiral Path of a Dog \(A\) dog is tied to a circular tree trunk of radius 1 ft by a long leash. He has managed to wrap the entire leash around the tree while playing in the yard, and he finds himself at the point \((1,0)\) in the figure. Seeing a squirrel, he runs around the tree counterclockwise, keeping the leash taut while chasing the intruder. (a) Show that parametric equations for the dog's path (called an involute of a circle) are $$ x=\cos \theta+\theta \sin \theta \quad y=\sin \theta-\theta \cos \theta $$ [Hint: Note that the leash is always tangent to the tree, so OT is perpendicular to \(T D .\) I (b) Graph the path of the dog for \(0 \leq \theta \leq 4 \pi\)

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\(/ z_{1}\) . $$ z_{1}=-20, \quad z_{2}=\sqrt{3}+i $$

Convert the polar equation to rectangular coordinates. $$ \cos 2 \theta=1 $$

More Information in Parametric Equations In this section we stated that parametric equations contain more information than just the shape of a curve. Write a short paragraph explaining this statement. Use the following example and your answers to parts (a) and (b) below in your explanation. The position of a particle is given by the parametric equations $$ x=\sin t \quad y=\cos t $$ where \(t\) represents time. We know that the shape of the path of the particle is a circle. (a) How long does it take the particle to go once around the circle? Find parametric equations if the particle moves twice as fast around the circle. (b) Does the particle travel clockwise or counterclockwise around the circle? Find parametric equations if the particle moves in the opposite direction around the circle.

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\(/ z_{1}\) . $$ z_{1}=3+4 i, \quad z_{2}=2-2 i $$

The Distance Formula in Polar Coordinates (a) Use the Law of Cosines to prove that the distance between the polar points \(\left(r_{1}, \theta_{1}\right)\) and \(\left(r_{2}, \theta_{2}\right)\) is $$ d=\sqrt{r_{1}^{2}+r_{2}^{2}-2 r_{1} r_{2} \cos \left(\theta_{2}-\theta_{1}\right)} $$ (b) Find the distance between the points whose polar coordinates are \((3,3 \pi / 4)\) and \((1,7 \pi / 6),\) using the formula from part (a). (c) Now convert the points in part (b) to rectangular coordinates. Find the distance between them using the usual Distance Formula. Do you get the same answer?

Find the indicated power using De Moivre's Theorem. $$ (1+i)^{20} $$

Find the indicated power using De Moivre's Theorem. $$ (1-\sqrt{3} i)^{5} $$

Find the indicated power using De Moivre's Theorem. $$ (2 \sqrt{3}+2 i)^{5} $$

Find the indicated power using De Moivre's Theorem. $$ (1-i)^{8} $$

Find the indicated power using De Moivre's Theorem. $$ \left(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i\right)^{12} $$

Find the indicated power using De Moivre's Theorem. $$ (\sqrt{3}-i)^{-10} $$

Find the indicated power using De Moivre's Theorem. $$ (2-2 i)^{8} $$

Find the indicated power using De Moivre's Theorem. $$ \left(-\frac{1}{2}-\frac{\sqrt{3}}{2} i\right)^{15} $$

Find the indicated power using De Moivre's Theorem. $$ (-1-i)^{7} $$

Find the indicated power using De Moivre's Theorem. $$ (2 \sqrt{3}+2 i)^{-5} $$

Find the indicated power using De Moivre's Theorem. $$ (1-i)^{-8} $$

Find the indicated roots, and graph the roots in the complex plane. The square roots of \(4 \sqrt{3}+4 i\)

Find the indicated roots, and graph the roots in the complex plane. The cube roots of \(4 \sqrt{3}+4 i\)

Find the indicated roots, and graph the roots in the complex plane. The fourth roots of \(-81 i\)

Find the indicated roots, and graph the roots in the complex plane. The fifth roots of 32

Find the indicated roots, and graph the roots in the complex plane. The eighth roots of 1

Find the indicated roots, and graph the roots in the complex plane. The cube roots of \(1+i\)

Find the indicated roots, and graph the roots in the complex plane. The cube roots of \(i\)

Find the indicated roots, and graph the roots in the complex plane. The fifth roots of \(i\)

Find the indicated roots, and graph the roots in the complex plane. The fourth roots of \(-1\)

Find the indicated roots, and graph the roots in the complex plane. The fifth roots of \(-16-16 \sqrt{3} i\)

Solve the equation. $$ z^{4}+1=0 $$

Solve the equation. $$ z^{8}-i=0 $$

Solve the equation. $$ z^{3}-4 \sqrt{3}-4 i=0 $$

Solve the equation. $$ z^{6}-1=0 $$

Solve the equation. $$ z^{3}+1=-i $$

Solve the equation. $$ z^{3}-1=0 $$

(a) Let \(w=\cos \frac{2 \pi}{n}+i \sin \frac{2 \pi}{n}\) where \(n\) is a positive integer. Show that \(1, w, w^{2}, w^{3}, \ldots, w^{n-1}\) are the \(n\) distinct nth roots of 1 . (b) If \(z \neq 0\) is any complex number and \(s^{n}=z,\) show that the \(n\) distinct \(n\) th roots of \(z\) are $$ s, s w, s w^{2}, s w^{3}, \ldots, s w^{n-1} $$

Products of Roots of Unity Find the product of the three cube roots of 1 (see Exercise \(97 .\) Do the same for the fourth, fifth, sixth, and eighth roots of \(1 .\) What do you think is the product of the \(n\) th roots of 1 for any \(n\) ?

Complex Coefficients and the Quadratic Formula The quadratic formula works whether the coefficients of the equation are real or complex. Solve these equations using the quadratic formula and, if necessary, De Moivre's Theorem. $$ \begin{array}{l}{\text { (a) } z^{2}+(1+i) z+i=0} \\ {\text { (b) } z^{2}-i z+1=0} \\ {\text { (c) } z^{2}-(2-i) z-\frac{1}{4} i=0}\end{array} $$