Question

If \( z = \frac{3i}{2} \), what is the value of \( \arg(z) \)?

• A. \(0 \)
• B. \( \frac{\pi}{2} \)
• C. \( \pi \)
• D. \( \frac{3\pi}{2} \)

Hint:

Write the number in the form $x+iy$ first, then find its modulus $r=\sqrt{x^2+y^2}$. Use both $\cos\theta = x/r$ and $\sin\theta = y/r$ together to pin down the exact angle, rather than checking only the sign of the imaginary part.

Answer 1

The Correct Option is B

Concept: For a complex number \( z = x + iy \), the argument \( \arg(z) \) is the angle made with the positive real axis. For a purely imaginary number:
• If \( y > 0 \), then \( \arg(z) = \frac{\pi}{2} \)
• If \( y < 0 \), then \( \arg(z) = -\frac{\pi}{2} \)

Step 1: Express the complex number.
Given: \[ z = \frac{3i}{2} = 0 + \frac{3}{2}i \]

Step 2: Identify its position in the complex plane.
Since the real part is \(0\) and the imaginary part is positive, the number lies on the positive imaginary axis.

Step 3: Determine the argument.
For any positive purely imaginary number: \[ \arg(z) = \frac{\pi}{2} \]

Answer 2

Concept:

• Any complex number $z = x + iy$ can be written in polar form as $z = r(\cos\theta + i\sin\theta)$, where $r = |z|$ is the modulus and $\theta = \arg(z)$ satisfies $\cos\theta = x/r$ and $\sin\theta = y/r$.
• Once both $\cos\theta$ and $\sin\theta$ are known exactly, the angle can be read off directly from standard values, without relying on separate rules for each axis.

Step 1: Write $z$ in the form $x + iy$ and find its modulus.
$z = \dfrac{3i}{2} = 0 + \dfrac{3}{2}i$, so $x = 0$ and $y = \dfrac{3}{2}$.
$r = \sqrt{x^2 + y^2} = \sqrt{0^2 + \left(\dfrac{3}{2}\right)^2} = \dfrac{3}{2}$

Step 2: Find $\cos\theta$ and $\sin\theta$.
$\cos\theta = \dfrac{x}{r} = \dfrac{0}{3/2} = 0$
$\sin\theta = \dfrac{y}{r} = \dfrac{3/2}{3/2} = 1$

Step 3: Identify $\theta$ from these standard values.
$\cos\theta = 0$ and $\sin\theta = 1$ together correspond to exactly one angle in the principal range $(-\pi, \pi]$.
$\theta = \dfrac{\pi}{2}$

Final Answer: $\arg(z) = \dfrac{\pi}{2}$