Answer the following questions about complex numbers a write

Answer the following questions about complex numbers: a. write 3(cos(330 degree) + i sin(330 degree)) in rectangular form. b. Write 7 + 7i Squareroot 3 in trigonometric form r(cos(theta) + i sin(theta)). Recall that theta must be in [0 degree, 360 degree). c. Compute (7 + 7i Squareroot 3)^20.

Solution

a     3(cos 330 + i sin 330)

x= 3 cos 330       and     y= 3sin 330

x=3(sqrt3)/2        and      y= -3/2

Therefore the required rectangular coordinate is (3(sqrt3)/2, -3/2)

b. 7 + 7isqrt3

   r= sqrt(a²+b²) = sqrt(7²+ (7sqrt3)²) = sqrt(196)=14

tan theta=y/x= 7sqrt3/7

theta=tan^-1sqrt3 = pi/3

Therefore the required coordinate is 14(cos (pi/3) + i sin(pi/3))

c. (7+7i sqrt3)²⁰

       from part b,we get (7 + 7isqrt3) =14(cos(pi/3) + i sin(pi/3))

Therefore (7 + 7isqrt3)²⁰ = (14(cos (pi/3) + i sin(pi/3)))²⁰

According to Demoivre\'s formula which is

(cos n + i sin n)^r = cos(r n) + i sin (r n)

Therefore (14(cos(pi/3) + i sin(pi/3)))²⁰= 14²⁰(cos (20pi/3) + i sin (20pi/3))

                                                             = 14²⁰(-1/2 + i (sqrt3)/2)