Let Z=a+i b ; Z=-7+ 3 i ; a=-7 ,b = 3Z=a+ib;Z=−7+3i;a=−7,b=3 ;
Z=-7+ 3 iZ=−7+3i is in 22 nd quadrant.
Modulus |Z|=sqrt(a^2+b^2)=(sqrt((-7)^2+ 3^2)) =sqrt 58 |Z|=√a2+b2=(√(−7)2+32)=√58
tan alpha =|b/a|= 3/7 or alpha =tan^-1(3/7) ~~ 0.4049tanα=∣∣∣ba∣∣∣=37orα=tan−1(37)≈0.4049
thetaθ is on 22 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]
