Find all the square roots of the complex number 14i Write th

Find all the square roots of the complex number 14i. Write the square roots in polar (re^i) form, with the smaller angle first. Give angles in degress

Solution

this is the polar form  r(cos()+isin())

we compare it with 1-4i

rcos =1

rsin=-4

squaring and adding them

r^2(cos^2 +sin^2) = 1 +(-4)^2

r^2 = 1+16

r= sqrt(17) =4.1

tan = -4/1

= arc tan(-4)

= -76 degree

square root of r(cos()+isin()) is ±r(cos(/2)+isin(/2)).

square root form is  ±(4.1) (cos(-76/2) + isin(-76/2)

  ±4.1(cos38 -isin38)

in polar form 4.1 e^(-38i)