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

Answer

N/a

Explanation

Let `z= (i- 1)/(cos""(pi)/(3)+isin""(pi)/(3))`
`" "=(i-1)/((1)/(2)+(isqrt(3))/(2))=(2(i-1 ))/((1+isqrt(3)))xx(1-isqrt(3))/(1-isqrt(3))`
`" "=(2(i-i^(2)sqrt(3)-1+isqrt(3)))/(1- 3i^(2))`
`" "=(2[ (sqrt(3)-1)+i(sqrt(3)+1)])/( 4)`
`" "=(sqrt(3)-1)/(2)+(i(sqrt(3)+1))/(2)`
Let `r(costheta+ isintheta)=(sqrt(3)-1)/(2)+(i(sqrt(3) +1))/( 2)`
` rArr" "rcostheta=(sqrt(3) -1)/(2) and rsin theta=(sqrt(3)+1)/(2)`
`therefore" "r^(2)=((sqrt(3)-1)/ (2))^(2)+((sqrt(3)+1)/(2))^(2)`
` " "=(8)/(4)=2`
`rArr" "r=sqrt(2)`
and ` " "tantheta=(sqrt(3)+1)/(sqrt(3)-1)=(1+(1)/(sqrt(3)))/(1-(1)/(sqrt3))`
`rArr" "tantheta=tan(45^(@)+30^(@))`
` " "=tan""(5pi)/(12)`
`rArr" "theta=(5pi)/( 12)`
`therefore` Polar form ` =sqrt(2)(cos""(5pi)/(12)+isin""(5pi)/(12)) `.