91Ó°ÊÓ

Write the complex number in polar form, \(r\) cis \(\theta\). $$1+i \sqrt{3}$$

Write the complex number in polar form, \(r\) cis \(\theta\). $$-3+4 i$$

Write the complex number in polar form, \(r\) cis \(\theta\). $$-8$$

Write the complex number in Cartesian form, \(a+b i\). $$6 \operatorname{cis} 120^{\circ}$$

Hyperbola Problem 2: Consider the polar equation \(r=\frac{8}{3+5 \cos \theta}.\) a. Plot the graph. Sketch the result. b. Show algebraically that the graph is a hyperbola by transforming the equation to Cartesian form. c. Where is one focus of the hyperbola? What is the eccentricity?

Write the complex number in Cartesian form, \(a+b i\). $$2 \text { cis } 0^{\circ}$$

Rose Problem: The general equation of a rose in polar coordinates is $$r=k \cos n \theta$$ where \(n\) is an integer. a. Plot the four-leaved rose \(r=9 \cos 2 \theta\) b. Plot the three-leaved rose \(r=9 \cos 3 \theta\) c. What is the relationship between \(n\) and the number of leaves in the rose? You may want to plot other roses to confirm your conclusion. d. Plot the five-leaved rose that appears in the icon at the top of even- numbered pages in this chapter. Make the distance from the pole to the tip of a leaf equal to 7 units.

Find a. \(z_{1} z_{2}\) b. \(\frac{z_{1}}{z_{2}}\) d. \(z_{2}^{3}\) C. \(z_{1}^{2}\) $$z_{1}=4 \operatorname{cis} 238^{\circ}, z_{2}=2 \operatorname{cis} 51^{\circ}$$

Find the indicated roots and sketch the answers on the complex plane. Cube roots of 8