Chapter 9

Sketch a graph of the polar equation, and express the equation in rectangular coordinates. $$ r=2 $$

Sketch the complex number \(z\) and its complex conjugate \(z\) on the same complex plane. $$ z=-5+6 i $$

\(3-24=A\) pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular coordinate equation for the curve by eliminating the parameter. $$ x=\cos 2 t, \quad y=\sin 2 t $$

Sketch a graph of the polar equation, and express the equation in rectangular coordinates. $$ r=-1 $$

Sketch \(z_{1}, z_{2}, z_{1}+z_{2},\) and \(z_{1} z_{2}\) on the same complex plane. $$ z_{1}=2-i, \quad z_{2}=2+i $$

\(3-24=A\) pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular coordinate equation for the curve by eliminating the parameter. $$ x=\sec t, \quad y=\tan t, \quad 0 \leq t<\pi / 2 $$

Sketch \(z_{1}, z_{2}, z_{1}+z_{2},\) and \(z_{1} z_{2}\) on the same complex plane. $$ z_{1}=-1+i, \quad z_{2}=2-3 i $$

\(3-24=A\) pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular coordinate equation for the curve by eliminating the parameter. $$ x=\tan t, \quad y=\cot t, \quad 0 < t < \pi / 2 $$

Sketch a graph of the polar equation, and express the equation in rectangular coordinates. $$ \theta=5 \pi / 6 $$

Sketch the set in the complex plane. $$ \\{z=a+b i | a \leq 0, b \geq 0\\} $$

\(3-24=A\) pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular coordinate equation for the curve by eliminating the parameter. $$ x=\tan t, \quad y=\cot t, \quad 0 < t < \pi / 2 $$

Sketch a graph of the polar equation, and express the equation in rectangular coordinates. $$ r=6 \sin \theta $$

Sketch the set in the complex plane. $$ \\{z=a+b i | a>1, b>1\\} $$

\(3-24=A\) pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular coordinate equation for the curve by eliminating the parameter. $$ x=\sec t, \quad y=\tan ^{2} t, \quad 0 \leq t<\pi / 2 $$

Sketch a graph of the polar equation, and express the equation in rectangular coordinates. $$ r=\cos \theta $$

Sketch the set in the complex plane. $$ \\{z| | z |=3\\} $$

\(3-24=A\) pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular coordinate equation for the curve by eliminating the parameter. $$ x=\cos ^{2} t, \quad y=\sin ^{2} t $$

Sketch a graph of the polar equation. $$ r=-2 \cos \theta $$

Sketch the set in the complex plane. $$ \\{z| | z | \geq 1\\} $$

\(3-24=A\) pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular coordinate equation for the curve by eliminating the parameter. $$ x=\cos ^{2} t, \quad y=\sin ^{2} t $$

Sketch a graph of the polar equation. $$ r=2 \sin \theta+2 \cos \theta $$

Sketch the set in the complex plane. $$ \\{z| | z |<2\\} $$

\(25-28\) . The position of an object in circular motion is modeled by the given parametric equations. Describe the path of the object by stating the radius of the circle, the position at time \(t=0\) , the orientation of the motion (clockwise or counterclockwise), and the time \(t\) that it takes to complete one revolution around the circle. $$ x=3 \cos t, \quad y=3 \sin t $$

Sketch a graph of the polar equation. $$ r=2-2 \cos \theta $$

Sketch the set in the complex plane. $$ \\{z|2 \leq| z | \leq 5\\} $$

\(25-28\) . The position of an object in circular motion is modeled by the given parametric equations. Describe the path of the object by stating the radius of the circle, the position at time \(t=0\) , the orientation of the motion (clockwise or counterclockwise), and the time \(t\) that it takes to complete one revolution around the circle. $$ x=2 \sin t, \quad y=2 \cos t $$

Sketch a graph of the polar equation. $$ r=1+\sin \theta $$

Find the rectangular coordinates for the point whose polar coordinates are given. $$ (4, \pi / 6) $$

Sketch a graph of the polar equation. $$ r=-3(1+\sin \theta) $$

Find the rectangular coordinates for the point whose polar coordinates are given. $$ (6,2 \pi / 3) $$

Sketch the set in the complex plane. $$ \\{z=a+b i | a \geq b\\} $$

\(25-28\) . The position of an object in circular motion is modeled by the given parametric equations. Describe the path of the object by stating the radius of the circle, the position at time \(t=0\) , the orientation of the motion (clockwise or counterclockwise), and the time \(t\) that it takes to complete one revolution around the circle. $$ x=4 \cos 3 t, \quad y=4 \sin 3 t $$

Sketch a graph of the polar equation. $$ r=\cos \theta-1 $$

Find the rectangular coordinates for the point whose polar coordinates are given. $$ (\sqrt{2},-\pi / 4) $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ 1+i $$

\(29-34=\) Find parametric equations for the line with the given properties. Slope \(\frac{1}{2},\) passing through \((4,-1)\)

Sketch a graph of the polar equation. $$ r=\sin 2 \theta $$

Find the rectangular coordinates for the point whose polar coordinates are given. $$ (-1,5 \pi / 2) $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ 1+\sqrt{3} i $$

\(29-34=\) Find parametric equations for the line with the given properties. Slope \(-2,\) passing through \((-10,-20)\)

Sketch a graph of the polar equation. $$ r=2 \cos 3 \theta $$

Find the rectangular coordinates for the point whose polar coordinates are given. $$ (5,5 \pi) $$

Sketch a graph of the polar equation. $$ r=-\cos 5 \theta $$

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

\(29-34=\) Find parametric equations for the line with the given properties. Passing through \((6,7)\) and \((7,8)\)

Find the rectangular coordinates for the point whose polar coordinates are given. $$ (0,13 \pi) $$

Sketch a graph of the polar equation. $$ r=\sin 4 \theta $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ 1-i $$

\(29-34=\) Find parametric equations for the line with the given properties. Passing through \((12,7)\) and the origin

Find the rectangular coordinates for the point whose polar coordinates are given. $$ (6 \sqrt{2}, 11 \pi / 6) $$