Find the modulus and argument of each of the following complex numbers and hence express each of them in polar form: `((1-i))/(("cos"(pi)/(3)+"i sin"(pi)/(3)))`

Find the modulus and argument of each of the following complex numbers

1.	Find the modulus and argument of each of the following complex numbers and hence express each of them in polar form: `((1-i))/(("cos"(pi)/(3)+"i sin"(pi)/(3)))`
Answer» Correct Answer - `sqrt(2),(7pi)/(12),sqrt(2)("cos"(7pi)/(12)-"i sin"(7pi)/(12))` Given number `=(2(1-i))/((1+sqrt(3)i))xx((1-sqrt(3)i))/((1-sqrt(3)i))={((1-sqrt(3)))/(2)-((1+sqrt(3)))/(2)i}` `therefore" "\|z\|^(2)=((1-sqrt(3))^(2))/(4)+((1+sqrt(3))^(2))/(4)=(2(1+3))/(4)=2 rArr \|z\| = sqrt(2)`. `tan alpha=\|((-(1+sqrt(3)))/(2))/(((1+sqrt(3)))/(2))\|=((sqrt(3)+1))/((sqrt(3)-1))=((1+(1)/(sqrt(3))))/((1-(1)/(sqrt(3))))=("tan"(pi)/(4)+"tan"(pi)/(6))/(1-"tan"(pi)/(4)*"tan"(pi)/(6))` `=tan((pi)/(4)+(pi)/(6))="tan"(5pi)/(12)`. The given number z is represented by the point `P((-(sqrt(3)-1))/(2),(-(sqrt(3)+1))/(2))` So, it lies in the third quadrant. `therefore" "arg(z) = theta = -(pi- alpha)=-(pi-(5pi)/(12))=(-7pi)/(12)`. Hence, `z=sqrt(2){cos((-7pi)/(12))+i sin((-7pi)/(12))}`.