Argument Of Complex Numbers

Question

The principal argument of the complex number given by z = -1 + i times the square root of 3, expressed in radians within the standard interval, equals which value?

RankGuru · Accepted Answer

The principal argument of a complex number is the angle its position vector makes with the positive real axis, measured in (-π, π], a definition essential to JEE Advanced polar work. For z = -1 + i√3, the real part is negative and the imaginary part is positive, placing z in the second quadrant. The reference angle satisfies tan(θ) = |√3 / -1| = √3, giving a base angle of π/3. In the second quadrant the principal argument is π - π/3 = 2π/3. Option π/3 ignores the quadrant and treats z as first-quadrant. Option -2π/3 corresponds to the third quadrant, wrong sign of imaginary part. Option 5π/6 uses an incorrect reference angle. Hence arg(z) = 2π/3. Plausibility check: the modulus is √(1 + 3) = 2, so z = 2(cos 2π/3 + i sin 2π/3) = 2(-1/2 + i√3/2) = -1 + i√3, exactly reproducing z and confirming the argument.

Argument Of Complex Numbers

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