Question

Perform the following division and write the quotient in trigonometric form. Write the magnitude in exact form. Writethe argument in radians and round it to twodecimal places if necessary.-4 - 2i3 + 2iyea

Answer 1

First step is to simplify the expression

-4-2i

------

3+2i

Multiply by the complex conjugate

-4-2i ( 3-2i)

------ * ----------------

3+2i ( 3-2i)

(-4-2i) (3-2i)

------------------

3^2 +2^2

-12 -6i +8i -4

----------------------

9+4

-16+2i

--------

13

Now we can find the magnitude

z = sqrt( a^2 + b^2) where a = real part and b = imaginary part

z = sqrt( 256/139 + 4/139) = sqrt(260/139) = 2 sqrt(65)/13

theta = tan^-1 ( b/a) = tan^-1 ( -2/16) = tan^-1 (-1/8) = .12

z = 2 sqrt(65)/13 ( cos .12 + i sin .12)