caShow your work in converting the polar equation to a recta

ca)Show your work in converting the polar equation to a rectangular equation. Then use your knowledge of the rectangular equation to describe the graph, in detail. What is it? What are its main characteristics? r = 6 cos

b)

Use DeMoivre\'s Theorem to find the indicated power of the complex number. Write the answer in rectangular form.

c)Find all the complex fifth roots of -32. Write the final answers in polar form with the arguments in degrees.

Solution

a) r = 6cos

x = rcos

r = 6 cos(theta)
r^2 = 6r cos(theta)

x= r cos(theta)

r^2 = 6x
x^2+y^2 = 6x
x^2+y^2=6x
x^2-6x+y^2=0
x^2-6x+6 +y^2=6
(x-3)^2 + y^2=6
circle with center 3,0) and radius sqrt6

b) [ 1/3(cospi/15 + isinpi/15)]^5

As per De-Movries theoresm: z^n = r^n(cos(n*x) + isin(n*x) )

So,[ 1/3(cospi/15 + isinpi/15)]^5 = (1/3)^5 [ cos(5pi/15) +isin(5pi/15)]

= (cospi/3 +isinpi/3)/3^5

= 1/486 +i*sqrt3/486

= (1+isqrt3)/486