Video: Finding the Squares of Complex Numbers in Polar Form

If 𝑧 = 3(cos 45Β°+ 𝑖 sin 45Β°), what is 𝑧²?

04:16

Video Transcript

If 𝑧 is equal to three multiplied by cos of 45 degrees plus 𝑖 sin of 45 degrees, what is 𝑧 squared?

So here, we’ve been given a complex number 𝑧 which is in polar form. Polar form of a complex number is the form π‘Ÿ multiplied by cos of πœƒ plus 𝑖 sin of πœƒ, where π‘Ÿ is the magnitude or modulus of the complex number and πœƒ is its argument, also sometimes called its amplitude. We’re asked to work out 𝑧 squared which is the product of 𝑧 with itself. So we need to recall how to multiply two complex numbers which are in polar form.

We recall that if we have one complex number 𝑧 one which is equal to π‘Ÿ one multiplied by cos of πœƒ one plus 𝑖 sin of πœƒ one and the second complex number 𝑧 two which is equal to π‘Ÿ two multiplied by cos of πœƒ two plus 𝑖 sin of πœƒ two, then the product of these two complex numbers is given by π‘Ÿ one multiplied by π‘Ÿ two all multiplied by cos of πœƒ one plus πœƒ two plus 𝑖 sin of πœƒ one plus πœƒ two.

This may look a little bit complicated. But actually, what it’s telling us is that we need to multiply the moduli of our two complex numbers together to give the modulus of the product; that’s π‘Ÿ one π‘Ÿ two. And then, we need to add the arguments of our two complex numbers together πœƒ one plus πœƒ two to give the argument of the product. We’ll look at why this is in a moment.

In our complex number then, we know that the value of π‘Ÿ for both 𝑧 one and 𝑧 two because they’re both the same they’re both just 𝑧 is three and the value of πœƒ one and πœƒ two is 45 degrees. So applying this general result then, we have that 𝑧 squared is equal to three multiplied by three multiplied by cos of 45 degrees plus 45 degrees plus 𝑖 sin of 45 degrees plus 45 degrees. Three multiplied by three is equal to nine and 45 degrees plus 45 degrees is equal to 90 degrees.

So our product simplifies to nine multiplied by cos of 90 degrees plus 𝑖 sin of 90 degrees. We could simplify this further because cos of 90 degrees is equal to zero and sin of 90 degrees is equal to one. But we’ll leave our complex number 𝑧 squared in its polar form. Now, to see why this result is true, we need to consider our complex numbers 𝑧 one and 𝑧 two in a different form. We need to consider them in exponential form.

Here, 𝑧 one is equal to π‘Ÿ one 𝑒 to the power of 𝑖 πœƒ one and 𝑧 two is equal to π‘Ÿ two 𝑒 to the power of 𝑖 πœƒ two. When we multiply these two complex numbers together in exponential form, we get π‘Ÿ one 𝑒 to the 𝑖 πœƒ one multiplied by π‘Ÿ two 𝑒 to the 𝑖 πœƒ two. We can we reorder the different parts in this product to bring π‘Ÿ one and π‘Ÿ two next to each other at the front. So we now have π‘Ÿ one π‘Ÿ two multiplied by 𝑒 to the 𝑖 πœƒ one and 𝑒 to the 𝑖 πœƒ two.

But one of our laws of indices tells us that when we’re multiplying together two powers with the same base, in this case that’s 𝑒, we can add those powers together. So 𝑒 to the power of 𝑖 πœƒ one multiplied by 𝑒 to the power of 𝑖 πœƒ two is equals 𝑒 to the power of 𝑖 πœƒ one plus 𝑖 πœƒ two. We can factorise 𝑖 from this power and write the power as 𝑖 multiplied by πœƒ one plus πœƒ two. We see then that the product 𝑧 one 𝑧 two is a complex number which has a modulus of π‘Ÿ one π‘Ÿ two. That’s the product of the individual moduli and it has an argument of πœƒ one plus πœƒ two. That’s the sum of the individual arguments.

If we were to convert this number from exponential to polar form, then we would get 𝑧 one 𝑧 two is equal to π‘Ÿ one π‘Ÿ two multiplied by cos of πœƒ one plus πœƒ two plus 𝑖 sin of πœƒ one plus πœƒ two, which is the general result that we wrote down at the start of the question.

Using this result then, we found that 𝑧 squared is equal to nine multiplied by cos of 90 degrees plus 𝑖 sin of 90 degrees.

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