1.

Convert the complex number `(-16)/(1+isqrt(3))`into polar form.

Answer» `z = (-16)/(1+isqrt3) = (-16)/(1+isqrt3)**(1-isqrt3)/(1-isqrt3)`
`=(-16+16sqrt3i)/(1+3) = -4+4sqrt3i`
`:.-4+4sqrt3i = r(costheta+isintheta) `
`rcostheta = -4 and rsintheta = 4sqrt3`
`r^2(cos^2theta+sin^2theta) =16+48`
`r^2 = 64=> r = 8`
`:.costheta = -4/8 = -1/2 and sintheta = 4sqrt3/8 = sqrt3/2`
So, `theta = (2pi)/3`
so, polar form will be,
`z = 8(cos((2pi)/3)+isin((2pi)/3))`


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