Given that 𝑧 one is equal to two
cos 𝜋 by six plus 𝑖 sin 𝜋 by six and 𝑧 two is one over root three multiplied by
cos 𝜋 by three plus 𝑖 sin 𝜋 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 by six. And the argument of our second
complex number is 𝜋 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 by six plus 𝜋 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 by six
plus two 𝜋 by six, which is equal to three 𝜋 by six or simply one-half 𝜋, 𝜋 by
And we found the product of our two
complex numbers. It’s two root three over three
multiplied by cos of 𝜋 by two plus 𝑖 sin of 𝜋 by two.