Put 𝑧 equals five root three 𝑒 to the 𝜋 over three 𝑖 in algebraic form.
We’ve been given this complex number 𝑧 in exponential form. That’s the form 𝑧 equals 𝑟𝑒 to the 𝑖𝜃, where 𝑟 is the absolute value of the complex number and 𝜃 is its argument measured in radians. And we know another way that we can express a complex number is in algebraic form, that is, 𝑧 equals 𝑎 plus 𝑏𝑖, which is what we’re going to convert this to. But in order to do this, we’ll start by expressing 𝑧 in its polar form. That’s the form 𝑧 equals 𝑟 multiplied by cos 𝜃 plus 𝑖 sin of 𝜃.
But why are we doing this? Well, by writing it in this way, we can then distribute the parentheses to give us 𝑧 equals 𝑟 cos of 𝜃 add 𝑖𝑟 sin of 𝜃. And from here, 𝑟 cos of 𝜃 is the real part of the complex number, 𝑎, and 𝑟 sin of 𝜃 is the imaginary part of the complex number, 𝑏. And that gives us the algebraic form of 𝑧 equals 𝑎 plus 𝑏𝑖. So let’s begin by writing this complex number in its polar form. We can see just by inspection that the modulus of the complex number 𝑟 equals five root three and that the argument 𝜃 is equal to 𝜋 over three. So, using our values of 𝑟 and 𝜃 and the general polar form for a complex number, we have that our complex number can be written as 𝑧 equals five root three multiplied by cos of 𝜋 over three plus 𝑖 sin of 𝜋 over three.
But let’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 𝑧 equals five root three multiplied by one over two add root three over two 𝑖. Distributing the parentheses then gives us 𝑧 equals five root three over two add five root three multiplied by root three over two 𝑖. 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 𝑧 equals five root three over two add 15 over two 𝑖.
So, by taking a complex number in exponential form and working out the value of 𝑟 and 𝜃 then converting the complex number into polar form, we were then able to convert it into algebraic form.