Video: Multiplying Complex Numbers in Polar Form

Given that 𝑧₁ = 2(cos (πœ‹/6) + 𝑖 sin (πœ‹/6)) and 𝑧₂ = (1/√3)(cos (πœ‹/3) + 𝑖 sin (πœ‹/3)), find 𝑧₁𝑧₂.

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Video Transcript

Given that 𝑧 one is equal to two cos πœ‹ by six plus 𝑖 sin πœ‹ by six and 𝑧 two is one over root three multiplied by cos πœ‹ by three plus 𝑖 sin πœ‹ by three, find 𝑧 one 𝑧 two.

We have been given two complex numbers represented in polar or trigonometric form. And we’re being asked to find their product. To multiply two complex numbers in polar form, we need to multiply their moduli and add their arguments.

The modulus of our first complex number is two. And the modulus of our second complex number is one over root three. Then, the argument of our first complex number is πœ‹ by six. And the argument of our second complex number is πœ‹ by three. We said we need to begin by multiplying the moduli of our two complex numbers. That’s two multiplied by one over root three, which is equal to two over root three.

Now at this stage before we move on to finding the argument of the product, we’re going to rationalize the denominator of this fraction. And we do that by multiplying by the numerator and the denominator of the fraction by root three. Two multiplied by root three is two root three. And root three multiplied by root three is simply three. So the modulus of the products of our complex numbers is two root three over three.

We’re then going to add their arguments. That’s πœ‹ by six plus πœ‹ by three. This time we’re going to add these by creating a common denominator. Here, that’s six. So we multiply the numerator and the denominator of our second fraction by two. And we see this becomes πœ‹ by six plus two πœ‹ by six, which is equal to three πœ‹ by six or simply one-half πœ‹, πœ‹ by two.

And we found the product of our two complex numbers. It’s two root three over three multiplied by cos of πœ‹ by two plus 𝑖 sin of πœ‹ by two.

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