Stewart Precalc 6e Chapter 8 Test

*(a)* Convert the point whose polar coordinates are \(\left(8, \dfrac{5 \pi}{4}\right)\) to rectangular coordinates. *(b)* Find two polar coordinate representations for the rectangular coordinate point \((- 6, 2 \sqrt{3})\), one with \(r > 0\) and one with \(r < 0\) and both with \(0 \leq \theta < 2 \pi\).

*(a)* Graph the polar equation \(r = 8 \cos \theta\). What type of curve is this? *(b)* Convert the equation to rectangular coordinates.

Graph the polar equation \(r = 3 + 6 \sin \theta\). What type of curve is this?

Let \(z = 1 + \sqrt{3} i\). *(a)* Graph \(z\) in the complex plane. *(b)* Write \(z\) in polar form. *(c)* Find the complex number \(z^9\).

Let \(z_1 = 4\left(\cos \dfrac{7 \pi}{12} + i \sin \dfrac{7 \pi}{12}\right)\) and \(z_2 = 2\left(\cos \dfrac{5 \pi}{12} + i \sin \dfrac{5 \pi}{12}\right)\). Find \(z_1 z_2\) and \(\dfrac{z_1}{z_2}\).

Find the cube roots of \(27 i\), and sketch these roots in the complex plane.

*(a)* Sketch the graph of the parametric curve \(x = 3 \sin t + 3, \quad y = 2 \cos t, \quad (0 \leq t \leq \pi)\) *(b)* Eliminate the parameter \(t\) in part (a) to obtain an equation for this curve in rectangular coordinates.

Find parametric equations for the line of slope \(2\) that passes through the point \((3, 5)\).