### Video Transcript

Given that the modulus of π sub
one is equal to two, where the principal argument of π sub one is equal to six π
plus five π, and the modulus of π sub two is equal to six, where the principal
argument of π sub two is equal to six π plus four π, find π sub one multiplied
by π sub two.

In this question, weβre given some
information about two complex numbers. In both cases, weβre given the
modulus and principal arguments of these two complex numbers. We need to use this to determine
the product of the two complex numbers. And to do this, we first recall the
principal argument of a complex number is an argument of the complex number where we
restrict it to be between negative π and π. So in particular, this is just one
possible argument of the complex numbers. And therefore, since weβre given
the moduli of these complex numbers and their arguments, we can write these numbers
in polar form. And this is useful because then we
can use the properties of multiplying complex numbers written in polar form to find
their product.

So to do this, letβs start by
recalling how we write a complex number in polar form. The polar form of a complex number
π is the form π multiplied by the cos of π plus π sin of π. The value of π is the modulus of
π and the value of π is the argument of π. And we can use this to write both
π sub one and π sub two in polar form. Letβs start with π sub one. First, we know the modulus of π
sub one is equal to two. And weβre told an argument of π
sub one is six π plus five π. Therefore, we can write π sub one
as two multiplied by the cos of six π plus five π plus π sin of six π plus five
π. We can do exactly the same for π
sub two. Its modulus is six and its
principal argument is six π plus four π. So, π sub two is equal to six
times the cos of six π plus four π plus π sin of six π plus four π.

And now weβre almost ready to
answer our question. Letβs recall how we multiply two
complex numbers given in polar form. If we have two complex numbers
given in polar form π and π, then we recall we can multiply π by π by
multiplying the moduli of these two numbers and adding their arguments. In this case, π times π is π
multiplied by π all multiplied by the cos of π plus π plus π sin of π plus
π. We multiply their moduli and we add
their arguments.

We can therefore use this to
multiply our two complex numbers, π sub one and π sub two. We need to multiply their
moduli. Two times six is 12. And we need to add their
arguments. Six π plus five π plus six π
plus four π is 12π plus nine π. We can then apply this to get our
final answer. π sub one times π sub two is 12
multiplied by the cos of 12π plus nine π plus π sin of 12π plus nine π.