What is the modulus r and principal argument theta of the complex number z = -sqrt(3) + i?
Answer and explanation
Correct answer: A. r = 2, theta = 5*pi/6 (or 150°)
- Step 1: Compute modulus: r = |z| = sqrt((-sqrt(3))^2 + 1^2) = sqrt(3 + 1) = sqrt(4) = 2.
- Step 2: Determine argument: z lies in the second quadrant (Re(z) < 0, Im(z) > 0).
- Step 3: The reference angle is alpha = arctan(|1 / (-sqrt(3))|) = arctan(1/sqrt(3)) = pi/6.
- Step 4: In Quadrant II: theta = pi - alpha = pi - pi/6 = 5*pi/6.
Answer
r = 2, theta = 5*pi/6 (or 150°).
Why not the others: 'r = 2, theta = pi/6' is in Quadrant I (positive real part); 'r = 4' forgets to take the square root; 'r = 2, theta = 7*pi/6' is in Quadrant III.
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