Chapter 9

In \(1-6\) plot each number in the complex plane. \(2+i\) and its complex conjugate \(2-i\) and their sum and product

In \(1-6\) draw the curve and find the area inside. \(r=1+\cos \theta\)

Find the polar coordinates \(r \geqslant 0\) and \(0 \leqslant 0<2 \pi\) of these points. $$ (x, y)=(0,1) $$

Plot each number in the complex plane. \(1+i\) and its square \((1+i)^{2}\) and its reciprocal \(1 /(1+i)\)

In \(1-6\) draw the curve and find the area inside. \(r=\sin \theta+\cos \theta\) from 0 to \(\pi\)

Convert to \(xy\) coordinates to draw and identify these curves. $$ r(\cos \theta-\sin \theta)=2 $$

Find the polar coordinates \(r \geqslant 0\) and \(0 \leqslant 0<2 \pi\) of these points. $$ (x, y)=(-4,0) $$

In \(1-6\) draw the curve and find the area inside. \(r=2+\cos \theta\)

Convert to \(xy\) coordinates to draw and identify these curves. $$ r=2 \cos \theta $$

Find the polar coordinates \(r \geqslant 0\) and \(0 \leqslant 0<2 \pi\) of these points. $$ (x, y)=(\sqrt{2}, \sqrt{2}) $$

Plot each number in the complex plane. The sixth roots of 1 (six of them)

Convert to \(xy\) coordinates to draw and identify these curves. $$ r=-2 \sin \theta $$

Find the polar coordinates \(r \geqslant 0\) and \(0 \leqslant 0<2 \pi\) of these points. $$ (x, y)=(-1, \sqrt{3}) $$

Plot each number in the complex plane. \(\cos 3 \pi / 4+i \sin 3 \pi / 4\) and its square and cube

Convert to \(xy\) coordinates to draw and identify these curves. $$ r=1 /(2+\cos \theta) $$

Find the polar coordinates \(r \geqslant 0\) and \(0 \leqslant 0<2 \pi\) of these points. $$ (x, y)=(3,4) $$

Plot each number in the complex plane. \(4 e^{i x / 3}\) and its square roots

Convert to \(xy\) coordinates to draw and identify these curves. $$ r=1 /(1+2 \cos \theta) $$

Find the polar coordinates \(r \geqslant 0\) and \(0 \leqslant 0<2 \pi\) of these points. $$ (x, y)=(3,4) $$

For complex numbers \(c=x+i y=r e^{i \theta}\) and their conjugates \(\bar{c}=x-i y=r e^{-1 f},\) find all possible locations in the complex plane of (1) \(c+\bar{c}\) (2) \(c-\bar{c}\) (3) \(c \bar{c}\) (4) \(c / \bar{c}\)

Sketch the curve and check for \(x, y,\) and \(r\) symmetry. \(r^{2}=4 \cos 2 \theta \quad\) (lemniscate)

Find rectangular coordinates \((x, y)\) from polar coordinates. $$ (r, \theta)=(2, \pi / 2) $$

Find \(x\) and \(y\) for the complex numbers \(x+i y\) at angles \(\theta=45^{\circ}, 90^{\circ}, 135^{\circ}\) on the unit circle. Verify directly that the square of the first is the second and the cube of the first is the third.

Sketch the curve and check for \(x, y,\) and \(r\) symmetry. \(r^{2}=4 \sin 2 \theta \quad\) (lemniscate)

Find rectangular coordinates \((x, y)\) from polar coordinates. $$ (r, \theta)=(1,3 \pi / 2) $$

If \(c=2+i\) and \(d=4+3 i\) find \(c d\) and \(c / d\). Verify that the absolute value \(|c d|\) equals \(|c|\) times \(|d|,\) and \(|c / d|\) equals \(|c|\) divided by \(|d|\)

Sketch the curve and check for \(x, y,\) and \(r\) symmetry. \(r=\cos 3 \theta \quad\) (three petals)

Find rectangular coordinates \((x, y)\) from polar coordinates. $$ (r, \theta)=(\sqrt{20}, \pi / 4) $$

Find a solution \(x\) to \(e^{i x}=i\) and a solution to \(e^{i x}=1 / e\) Then find a second solution.

Sketch the curve and check for \(x, y,\) and \(r\) symmetry. \(r^{2}=10+6 \cos 4 \theta\)

Find rectangular coordinates \((x, y)\) from polar coordinates. $$ (r, \theta)=(3 \pi, 3 \pi) $$

Sketch the curve and check for \(x, y,\) and \(r\) symmetry. \(r=e^{\theta}\)

Find rectangular coordinates \((x, y)\) from polar coordinates. $$ (r, \theta)=(2,-\pi / 6) $$

Find the sum and product of the numbers. \(e^{i \theta}\) and \(e^{i \phi},\) also \(e^{x i / 4}\) and \(e^{-\pi i / 4}\)

Find the area between the curves in \(7-12\) after locating their intersections (draw them first). circle \(r=10\) beyond line \(r \cos \theta=6\)

Sketch the curve and check for \(x, y,\) and \(r\) symmetry. \(r=1 / \theta \quad\) (hyperbolic spiral)

Find rectangular coordinates \((x, y)\) from polar coordinates. $$ (r, \theta)=(2,5 \pi / 6) $$

Find the sum and product of the numbers. The sixth roots of 1 (add and multiply all six)

Locate the mistake and find the correct area of the lemniscate \(r^{2}=\cos 2 \theta:\) area \(=\int_{0}^{\pi} \frac{1}{2} r^{2} d \theta=\int_{0}^{\pi} \frac{1}{2} \cos 2 \theta d \theta=0\).

Sketch the curve and check for \(x, y,\) and \(r\) symmetry. \(r=\tan \theta\)

What is the distance from \((x, y)=(\sqrt{3}, 1)\) to \((1,-\sqrt{3})\) ?

Find the sum and product of the numbers. The two roots of \(c^{2}-4 c+5=0\)

Sketch the curve and check for \(x, y,\) and \(r\) symmetry. \(r=1-2 \sin 3 \theta \quad\) (rose inside rose)

How far is the point \(r=3, \theta=\pi / 2\) from \(r=4, \theta=\pi ?\)

If \(c=r e^{i \theta}\) is not zero, what are \(c^{4}\) and \(c^{-1}\) and \(c^{-4}\) ?

Convert \(r=6 \sin \theta+8 \cos \theta\) to the \(x y\) equation of a circle (what radius, what center?).

How far is \((x, y)=(r \cos \theta, r \sin \theta)\) from \((X, Y)=(R \cos \phi\), \(R \sin \phi)\) ? Simplify \((x-X)^{2}+(y-Y)^{2}\) by using \(\cos (\theta-\phi)=\) \(\cos \theta \cos \phi+\sin \theta \sin \phi\)

Multiply out \((\cos \theta+i \sin \theta)^{3}=e^{i 3 \theta},\) to find the real part \(\cos 3 \theta\) and the imaginary part \(\sin 3 \theta\) in terms of \(\cos \theta\) and \(\sin \theta\)

Squaring and adding in the Mars-Earth equation gives \(x_{M-E}^{2}+y_{M-E}^{2}=13-12 \cos \pi t\). The graph of \(r^{2}=13-\) \(12 \cos \theta\) is not at all like Figure \(9.4 \mathrm{~d}\). What went wrong?

Find a second set of polar coordinates (a different \(r\) or \(\theta\) ) for the points $$ (r, \theta)=(-1, \pi / 2), \quad(-1,3 \pi / 4), \quad(1,-\pi / 2), $$