Let #Z=a+i b ; Z=-7+ 3 i ; a=-7 ,b = 3# ;
#Z=-7+ 3 i# is in #2# nd quadrant.
Modulus #|Z|=sqrt(a^2+b^2)=(sqrt((-7)^2+ 3^2)) =sqrt 58 #
# tan alpha =|b/a|= 3/7 or alpha =tan^-1(3/7) ~~ 0.4049#
#theta# is on #2# nd quadrant # :. theta=pi-0.4049#
# :. theta~~ 2.7367#. Argument , # theta ~~2.7367:. #
In trigonometric form expressed as
#r(cos theta+isintheta) = sqrt58(cos 2.74+i sin 2.74) #
#Z=1+ 3 i# is in #1# st quadrant.
Modulus #|Z|=sqrt(a^2+b^2)=(sqrt(1^2+ 3^2)) =sqrt 10 #
# tan alpha =|b/a|= 3/1 or alpha =tan^-1(3) ~~ 1.249#
#theta# is on #1# st quadrant # :. theta=1.249#
Argument , # theta ~~1.249:. #
In trigonometric form expressed as
#r(cos theta+isintheta) = sqrt 10 (cos 1.25+i sin 1.25) #
#(-7+ 3 i)(1+ 3 i) = #
# sqrt58(cos 2.74+i sin 2.74) * sqrt 10 (cos 1.25+i sin 1.25)~~ #
#sqrt58 * sqrt 10 ( cos (2.74+1.25) + i sin(2.74+1.25) ~~#
#24.08 ( cos (3.99) + i sin(3.99) = (-16-18 i)# [Ans]
