The principal argument of z = -2 - 2√3 i is which value?

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The principal argument of z = -2 - 2√3 i is which value?

The main idea is finding the principal argument, which is the angle θ in (-π, π] that describes the direction of the complex number in the plane. For z = -2 - 2√3 i, the point has coordinates (-2, -2√3), so it lies in the third quadrant. First compute the modulus: r = sqrt((-2)^2 + (-2√3)^2) = sqrt(4 + 12) = 4. Then the cosine and sine of θ are cos θ = x/r = -2/4 = -1/2 and sin θ = y/r = (-2√3)/4 = -√3/2. This corresponds to an angle of 4π/3 (240°) in standard position. But the principal argument must lie between -π and π, so subtract 2π: 4π/3 − 2π = -2π/3. Therefore, the principal argument is -2π/3.

The main idea is finding the principal argument, which is the angle θ in (-π, π] that describes the direction of the complex number in the plane. For z = -2 - 2√3 i, the point has coordinates (-2, -2√3), so it lies in the third quadrant.

First compute the modulus: r = sqrt((-2)^2 + (-2√3)^2) = sqrt(4 + 12) = 4. Then the cosine and sine of θ are cos θ = x/r = -2/4 = -1/2 and sin θ = y/r = (-2√3)/4 = -√3/2. This corresponds to an angle of 4π/3 (240°) in standard position.

But the principal argument must lie between -π and π, so subtract 2π: 4π/3 − 2π = -2π/3. Therefore, the principal argument is -2π/3.

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