91Ó°ÊÓ

The \(n\)th roots of a complex number are equally spaced on a circle of radius \(r\), since their arguments all differ by _________ degrees or _________ radians.

Write out the procedure for plotting points in polar coordinates, as though you were explaining the process to a friend.

Discuss the similarities between finding the components of a vector and writing a complex number in trigonometric form.

Use De Moivre's theorem to verify the solution given for each polynomial equation. $$ z^{5}+z^{4}-4 z^{3}-4 z^{2}+16 z+16=0 ; z=\sqrt{3}-i $$

Find the \(n\)th roots indicated by writing and solving the related equation. $$ \text { five fifth roots of } 243 $$

Use the nth roots theorem to find the \(n\)th roots. Clearly state \(r, n\), and \(\theta\) (from the trigonometric form of \(z\) ) as you begin. Answer in exact form when possible, otherwise use a four decimal place approximation. $$ \text { four fourth roots of }-7-7 i $$

The Folium of Descartes: $$ x(t)=\frac{3 k t}{1+t^{3}} ; y(t)=\frac{3 k t^{2}}{1+t^{3}} $$ The Folium of Descartes is a parametric curve developed by Descartes in order to test the ability of Fermat to find its maximum and minimum values. a. Graph the curve on a graphing calculator with \(k=1\) using a reduced window ( zoom 4), with Tmin \(=-6\), Tmax \(=6\), and Tstep \(=0.1\). Locate the coordinates of the tip of the folium (the loop). b. This graph actually has a discontinuity (a break in the graph). At what value of \(t\) does this occur? c. Experiment with different values of \(k\) and generalize its effect on the basic graph.

Equilateral triangles in the complex plane: $$ u^{2}+v^{2}+w^{2}=u v+u w+v w $$ If the line segments connecting the complex numbers \(u, v\), and \(w\) form the vertices of an equilateral triangle, the formula shown holds true. Verify that \(u=2+\sqrt{3} i, v=10+\sqrt{3} i\), and \(w=6+5 \sqrt{3} i\) form the vertices of an equilateral triangle using the distance formula, then verify the formula given.

Use a sum or difference identity to write the value of \(\cos 105^{\circ}\) in exact form.

The wheels on a motorcycle are rotating at \(1000 \mathrm{rpm}\). If they have a 12 -in. radius, how fast is the motorcycle traveling in miles per hour?