### 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 weβre 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. Weβll 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. Thatβs 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. Thatβs 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 thatβs left is to substitute these values into the general form of a complex
number in polar form.

Itβs eight cos of nine π minus five π plus π sin of nine π minus five π.