Question

Let \(Z_1 = r_1(\cos \theta_1 + i\sin \theta_1)\) and \(Z_2 = r_2(\cos \theta_2 + i\sin \theta_2)\) be two complex numbers, then value of \(Z_1Z_2\) is

• A. \(r_1r_2[\cos(\theta_1\theta_2) + i\sin(\theta_1\theta_2)]\)
• B. \(r_1r_2[\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)]\)
• C. \(r_1r_2[\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)]\)
• D. \(\frac{r_1}{r_2}[\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)]\)

Hint:

When multiplying complex numbers in polar form:
- Multiply their magnitudes: \( r_{\text{new}} = r_1 \cdot r_2 \).
- Add their arguments (angles): \( \theta_{\text{new}} = \theta_1 + \theta_2 \).
This simple rule saves you from performing long algebraic expansions.

Answer 1

The Correct Option is B