### Video Transcript

Given that the modulus of 𝑧 is
three and the argument of 𝑧 is 𝜃 which is 𝜋 over three, find 𝑧, giving your
answer in algebraic form.

When we write a complex number in
algebraic or rectangular form, we write it as 𝑧 is equal to 𝑎 plus 𝑏𝑖. We can then use these conversion
formulae to convert the polar coordinates with the modulus of 𝑟 and an argument of
𝜃 into the corresponding rectangular form. The modulus of our complex number
is three. And the argument is 𝜋 over
three.

So we can substitute these values
into the conversion formulae for 𝑎 and 𝑏. 𝑎 is equal to three multiplied by
cos 𝜋 over three, which is three over two. And 𝑏 is equal to three multiplied
by sin of 𝜋 over three, which is three root three over two. This means that the rectangular or
algebraic form of our complex number is three over two plus three root three over
two 𝑖.