1.

Convert the complex number `z=(i-1)/(cospi/3+isinpi/3)`in the polar form.

Answer» Given that, `z =(1-i)/("cos"(pi)/(3)+i " sin" (pi)/(3))=(-sqrt(2)[(-1)/(sqrt(2))+i(1)/sqrt(2)])/("cos"(pi)/(3)+i " sin" (pi)/(3)) `
`= (-sqrt(2)[cos(pi = pi//4) + i sin (pi - pi //4))]/(cos pi//3 + i sin pi//3)`
` = (-sqrt(2) [cos3pi//4 + i sin pi//4]) /(cos pi//3 + i sin pi//3)`
`=sqrt(2)[cos((3pi)/(4) - (pi)/(3)) + i sin ((3pi)/(4) - (pi)/(3))]`
`=sqrt(2)["cos"(5pi)/(12) + "i sin" (5pi)/(12)]`


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