# Question Video: Forming a Complex Number in Algebraic Form given Its Principal Argument and Modulus

Given that |𝑍| = 12 and the argument of 𝑍 is 𝜃 = 120°, find 𝑍, giving your answer in algebraic form.

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### Video Transcript

Given that the modulus of 𝑍 is 12 and the argument of 𝑍 is 𝜃 which is 120 degrees, find 𝑍, giving your answer in algebraic form.

When we represent a complex number in algebraic or rectangular form, we write it as 𝑎 plus 𝑏𝑖. We can use the following conversion formulae for converting the polar coordinates with the modulus of 𝑟 and an argument 𝜃 into its corresponding rectangular form. 𝑎 is equal to 𝑟 cos 𝜃. And 𝑏 is equal to 𝑟 sin 𝜃. We know that the modulus of our complex number is 12. And the argument 𝜃 is 120 degrees.

Let’s substitute these values into the conversion formulae for 𝑎 and 𝑏. 𝑎 is 12 multiplied by cos of 120 degrees. And as long as we make sure that our calculator is in degrees mode, that gives us a value of negative six. 𝑏 is 12 multiplied by sin of 120 degrees, which is six root three. This means that the rectangular or algebraic form of the complex number is negative six plus six root three 𝑖.