# Question Video: Multiplying Complex Numbers in Polar Form Mathematics

Given that π§β = 2(cos (5π β 2π) + π sin (5π β 2π)) and π§β = 4(cos(4π β 3π) + π sin(4π β 3π)), find π§βπ§β.

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### Video Transcript

Given that π§ one is equal to two cos of five π minus two π plus π sin of five π minus two π and π§ two equals four cos of four π minus three π plus π sin of four π minus three π, find π§ one multiplied by π§ two.

We have been given two complex numbers represented in polar or trigonometric form. And weβre looking to find their product. Remember, to multiply complex numbers in polar form, we multiply their moduli. And we add their arguments. And the general form of a complex number in polar form is π cos π plus π sin π, where π is the modulus and π is the argument. Weβll compare this general form to the complex numbers in our question.

The modulus of our first complex number π§ one is two. And the modulus of our second complex number is four. The argument of our first complex number is five π minus two π. And the argument of our second complex number is four π minus three π.

We said that, to find the modulus of the product of these two complex numbers, we need to find the product of their moduli. Thatβs two multiplied by four, which is of course eight. And we said that, to find the argument of π§ one multiplied by π§ two, we add their respective arguments. Thatβs five π minus two π plus four π minus three π.

We can collect like terms. And we see that the argument of the product of π§ one and π§ two is nine π minus five π. And all thatβs left is to substitute these values into the general form of a complex number in polar form.

Itβs eight cos of nine π minus five π plus π sin of nine π minus five π.