Chapter 9

A complex number \(z=a+b i\) has two parts: a is the _____ part, and \(b\) is the _____ part. To graph \(a+b i\) we graph the ordered pair \((\square , \square)\) in the complex plane.

Let \(z=a+b i\) (a) The modulus of \(z\) is r= _____, and an argument of \(z\) is an angle \(\theta\) satisfying \(\tan \theta=\) _____. (b) We can express \(z\) in polar form as \(z=\) _____ where \(r\) is the modulus of \(z\) and \(\theta\) is the argument of \(z\).

Plot the point that has the given polar coordinates. $$ (4, \pi / 4) $$

\(3-24=A\) 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. $$ x=2 t, \quad y=t+6 $$

Plot the point that has the given polar coordinates. $$ (1,0) $$

\(3-24=A\) 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. $$ x=6 t-4, \quad y=3 t, \quad t \geq 0 $$

Plot the point that has the given polar coordinates. $$ (6,-7 \pi / 6) $$

\(3-24=A\) 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. $$ x=t^{2}, \quad y=t-2, \quad 2 \leq t \leq 4 $$

Graph the complex number and find its modulus. $$ 4 i $$

Plot the point that has the given polar coordinates. $$ (3,-2 \pi / 3) $$

\(3-24=A\) 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. $$ x=2 t+1, \quad y=\left(t+\frac{1}{2}\right)^{2} $$

Graph the complex number and find its modulus. $$ -3 i $$

Plot the point that has the given polar coordinates. $$ (-2,4 \pi / 3) $$

\(3-24=A\) 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. $$ X=\sqrt{t}, \quad y=1-t $$

Graph the complex number and find its modulus. $$ -2 $$

Plot the point that has the given polar coordinates. $$ (-5,-17 \pi / 6) $$

Graph the complex number and find its modulus. $$ 6 $$

\(3-24=A\) 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. $$ x=t^{2}, \quad y=t^{4}+1 $$

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\) . $$ (3, \pi / 2) $$

\(3-24=A\) 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. $$ x=\frac{1}{t}, \quad y=t+1 $$

Graph the complex number and find its modulus. $$ 5+2 i $$

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2 .\) $$ r=2-\sin \theta $$

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\) . $$ (2,3 \pi / 4) $$

Graph the complex number and find its modulus. $$ 7-3 i $$

\(3-24=A\) 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. $$ x=t+1, \quad y=\frac{t}{t+1} $$

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2 .\) $$ r=4+8 \cos \theta $$

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\) . $$ (-1,7 \pi / 6) $$

Graph the complex number and find its modulus. $$ \sqrt{3}+i $$

\(3-24=A\) 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. $$ x=4 t^{2}, \quad y=8 t^{3} $$

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2 .\) $$ r=3 \sec \theta $$

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\) . $$ (-2,-\pi / 3) $$

Graph the complex number and find its modulus. $$ -1-\frac{\sqrt{3}}{3} i $$

\(3-24=A\) 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. $$ x=|t|, \quad y=|1-| t| | $$

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2 .\) $$ r=5 \cos \theta \csc \theta $$

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\) . $$ (-5,0) $$

Graph the complex number and find its modulus. $$ \frac{3+4 i}{5} $$

\(3-24=A\) 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. $$ x=2 \sin t, \quad y=2 \cos t, \quad 0 \leq t \leq \pi $$

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2 .\) $$ r=\frac{4}{3-2 \sin \theta} $$

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\) . $$ (3,1) $$

\(3-24=A\) 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. $$ x=2 \cos t, \quad y=3 \sin t, \quad 0 \leq t \leq 2 \pi $$

Graph the complex number and find its modulus. $$ \frac{-\sqrt{2}+i \sqrt{2}}{2} $$

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2 .\) $$ r=\frac{5}{1+3 \cos \theta} $$

Sketch the complex number \(z,\) and also sketch \(2 z,-z\) and \(\frac{1}{2} z\) on the same complex plane. $$ z=1+i $$

\(3-24=A\) 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. $$ x=\sin ^{2} t, \quad y=\sin ^{4} t $$

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2 .\) $$ r^{2}=4 \cos 2 \theta $$

Different Polar Coordinates for the Same Point: Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\). (3,1)

Sketch the complex number \(z,\) and also sketch \(2 z,-z\) and \(\frac{1}{2} z\) on the same complex plane. $$ z=-1+i \sqrt{3} $$

\(3-24=A\) 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. $$ X=\sin ^{2} t, \quad y=\cos t $$

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2 .\) $$ r^{2}=9 \sin \theta $$

Sketch the complex number \(z\) and its complex conjugate \(z\) on the same complex plane. $$ z=8+2 i $$