Problem 98 - Algebra and Trigonometry

Question

Explain how to find the product of two complex numbers in polar form.

Step-by-Step Solution

Answer

The product of two complex numbers in polar form can be found by multiplying their magnitudes and adding their angles.

Step 1: Understand Complex Numbers in Polar Form

A complex number can be represented in polar form as \( r(\cos \theta + i \sin \theta)\), where \( r\) is the magnitude of the complex number (also known as the modulus) and \( \theta \) is the angle it makes with the positive real axis (also known as the argument or phase).

Step 2: Formulation of Product Rule

The product of two complex numbers in polar form is given by multiplying their magnitudes and adding their angles. If the first complex number is \( r_1 (\cos \theta_1 + i \sin \theta_1)\) and the second is \( r_2 (\cos \theta_2 + i \sin \theta_2)\), then their product is \( r_1 r_2 (\cos(\theta_1+\theta_2) + i \sin(\theta_1+\theta_2))\).

Step 3: Apply the Product Rule

Use the product rule derived in the previous step to find the product of the two complex numbers in polar form. Multiply the magnitudes \( r_1 r_2 \) and add the angles \( \theta_1+\theta_2 \) to obtain the result.