Question Video: ο»ΏConverting Complex Numbers from Exponential to Algebraic Form Mathematics • 12th Grade

Put π§ = 5β(3)π^(π/3π) in algebraic form.

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Video Transcript

Put π§ equals five root three π to the π over three π in algebraic form.

Weβve been given this complex number π§ in exponential form. Thatβs the form π§ equals ππ to the ππ, where π is the absolute value of the complex number and π is its argument measured in radians. And we know another way that we can express a complex number is in algebraic form, that is, π§ equals π plus ππ, which is what weβre going to convert this to. But in order to do this, weβll start by expressing π§ in its polar form. Thatβs the form π§ equals π multiplied by cos π plus π sin of π.

But why are we doing this? Well, by writing it in this way, we can then distribute the parentheses to give us π§ equals π cos of π add ππ sin of π. And from here, π cos of π is the real part of the complex number, π, and π sin of π is the imaginary part of the complex number, π. And that gives us the algebraic form of π§ equals π plus ππ. So letβs begin by writing this complex number in its polar form. We can see just by inspection that the modulus of the complex number π equals five root three and that the argument π is equal to π over three. So, using our values of π and π and the general polar form for a complex number, we have that our complex number can be written as π§ equals five root three multiplied by cos of π over three plus π sin of π over three.

But letβs actually evaluate the values of cos of π over three and sin of π over three. Based on the unit circle, we can find that cos of π over three equals one over two, and we also get that sin of π over three equals root three over two. So, this gives us π§ equals five root three multiplied by one over two add root three over two π. Distributing the parentheses then gives us π§ equals five root three over two add five root three multiplied by root three over two π. But we know that root three multiplied by root three just gives us three and that five multiplied by three gives us 15 so that then gives us a final answer of π§ equals five root three over two add 15 over two π.

So, by taking a complex number in exponential form and working out the value of π and π then converting the complex number into polar form, we were then able to convert it into algebraic form.