Let zsqrt3i w22i 1 Convert z and w to polar form 2 Calculate

Let z=sqrt3-i w=-2-2i

1) Convert z and w to polar form

2) Calculate wz using De Moivre\'s theorem

3) Calculate z^2. Express your answer in rectangular (a+bi) form

4) Calculate w^4. Express your answer in polar (r(cos theta+i sin theta) form.

Solution

z = sqrt 3 - i

a = sqrt 3 , b = -1

r^2 = a^2 + b^2

r^2 = 3 + 1

r = 2

cos theta = a/r = sqrt 3 / 2

sin theta = b/r = -1 /2

therefore, polar form is

r ( cos theta + i sin theta )

z = 2 ( cos (-30) + i sin (-30) )

w = 2 - 2i

r^2 = 2^2 + 2^2

r = sqrt 8

theta = tan^-1 ( -2/-2)

theta = 45 degrees

polar form is

sqrt 8 ( cos 45 + i sin 45 )

2) wz = sqrt 8 ( cos 45 + i sin 45 )* 2 ( cos (-30) + i sin (-30) )

= 4 sqrt 2 e^( 45 - 30)i

= 4 sqrt 2 e^i15

3) z^2 = ( sqrt 3 - i )^2

r = 2

theta = -30 =-pi/6

z^2 = (2)^2 [ cos 2*(-pi/6) + i sin 2*(-pi/6)]

z^2 = 4 [ cos (-pi/3) + i sin (-pi/3) ]

z^2 = 4 [ 0.5 - i sqrt 3/2 ]

= 2 - 2i sqrt 3

4) w^4 = ( -2 - 2i)^4

w^4 = ( sqrt 8)^4 [ cos 4*(pi/4) + i sin (4*pi/4) ]

w^4 = 64 [ cos pi + i sin pi ]