Polar Form of Complex Numbers — Quiz

Question 4 • Precalculus

What is the polar form of negative 2 square root of 3 minus 6 i ?

Answer

2 square root of 6 open paren cosine of open paren 4 pi over 3 close paren plus i sine open paren 4 pi over 3 close paren close paren

2 square root of 6 open paren cosine of open paren 2 pi over 3 close paren plus i sine open paren 2 pi over 3 close paren close paren

4 square root of 3 open paren cosine of open paren 4 pi over 3 close paren plus i sine open paren 4 pi over 3 close paren close paren

4 square root of 3 open paren mathrm cosine open paren 2 pi over 3 close paren plus i sine open paren 2 pi over 3 close paren close paren

Explanation

For z = -2√3 - 6i, the modulus is √((-2√3)² + (-6)²) = √48 = 4√3. Both parts are negative, so the point is in quadrant III; its reference angle is π/3, giving θ = 4π/3. Thus the polar form is 4√3(cos(4π/3) + i sin(4π/3)), which is choice C.

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