91Ó°ÊÓ

For each equation, find an equivalent equation in rectangular coordinates, and graph. $$r=\frac{2}{1-\cos \theta}$$

Concept Check The complex number \(z,\) where \(z=x+y i,\) can be graphed in the plane as \((x, y) .\) Describe the graphs of all complex numbers z satisfying the conditions. The real and imaginary parts of \(z\) are equal.

For each equation, find an equivalent equation in rectangular coordinates, and graph. $$r=\frac{3}{1-\sin \theta}$$

A person in a plane flying straight north observes a mountain at a bearing of \(24.1^{\circ} .\) At that time, the plane is \(7.92 \mathrm{km}\) from the mountain. A short time later, the bearing to the mountain becomes \(32.7^{\circ} .\) How far is the airplane from the mountain when the second bearing is taken?

Given \(\mathbf{u}=\langle- 2,5\rangle\) and \(\mathbf{v}=\langle 4,3\rangle,\) find each of the following. $$\mathbf{v}-\mathbf{u}$$

Standing on one bank of a river flowing north, Mark notices a tree on the opposite bank at a bearing of \(115.45^{\circ} .\) Lisa is on the same bank as Mark, but \(428.3 \mathrm{m}\) away. She notices that the bearing of the tree is \(45.47^{\circ} .\) The two banks are parallel. What is the distance across the river?

Show that if \(z\) is an \(n\) th root of \(1,\) then so is \(\frac{1}{z}\)

Given \(\mathbf{u}=\langle- 2,5\rangle\) and \(\mathbf{v}=\langle 4,3\rangle,\) find each of the following. $$-4 \mathbf{u}$$

Concept Check The complex number \(z,\) where \(z=x+y i,\) can be graphed in the plane as \((x, y) .\) Describe the graphs of all complex numbers z satisfying the conditions. The real part of \(z\) is 1

Explain why a real number can have only one real cube root.