### Video Transcript

Given π§ is equal to two root three multiplied by cos of 240 degrees plus π sin of 240 degrees, find π§ squared in exponential form.

Weβre currently given a complex number written in trigonometric form. And weβre looking to find π§ squared in exponential form. There are two ways we can go about this. We can evaluate π§ squared in trigonometric form, and then convert it to exponential form. Or we can convert it to exponential form first, and then work out the value of π§ squared. Letβs consider both of these methods.

And to square this complex number, we recall De Moivreβs theorem. And this said, for a complex number in trigonometric form π cos π plus π sin π, this complex number to the power of π is given by π to the power of π multiplied by cos ππ plus π sin ππ. And in this example, π is a natural number.

We can see that the modulus of our complex number π§ is two root three. And π, its argument is 240 degrees. In fact, at some point, weβre going to have to convert this to radians. So, we might as well do that now and get it out of the way. To do this, we recall the fact that two π radians is equal to 360 degrees. And we can find the value of one degree by dividing through by 360. One degree is equal to two π over 360 radians. And two π by 360 simplifies to π by 180. So, one degree is equal to π over 180 radians. So, we can change 240 degrees into radians by multiplying it by π over 180. That gives us four π by three.

And so, we can work out the modulus of π§ squared by squaring the modulus of π§. Thatβs two root three squared. Root three squared is three. So, two root three squared is two squared multiplied by three, which is 12. And then, to work out the argument of π§ squared, we multiply the argument of π§ by the power thatβs two. Four π by three multiplied by two is eight π by three. So, we can see that in trigonometric form, π§ squared is 12 multiplied by cos of eight π over three plus π sin of eight π over three.

And remember, to change a complex number in trigonometric form into exponential form, itβs ππ to the ππ. And since π for π§ squared the modulus is 12 and the argument π is eight π by three, we can say that π§ squared is equal to 12π to the eight π over three π. Remember though, we usually want to represent this using the principal argument. Thatβs greater than negative π and less than or equal to π. In fact, eight π by three is greater than π. So, to find the principal argument, we add or subtract multiples of two π.

Here, letβs subtract two π from eight π by three. Two π is equal to six π over three. And when we subtract six π over three from eight π over three, weβre left with two π over three. So, in exponential form, π§ squared is 12π to the two π by three π.

Now, letβs consider the alternative method. And that was to convert this complex number into exponential form first and then square it. Once again, weβll use this rule. A complex number with a modulus of π and an argument π can be represented in exponential form as ππ to the ππ. We already solved 240 degrees is equal to four π by three radians. So, we can say that π§ in exponential form is two root three multiplied by π to the power of four π by three π.

And this time, to find π§ squared, we consider the alternative form of De Moivreβs theorem. And that says that if π§ is equal to ππ to the ππ, then π§ to the power of π is equal to π to the power of π multiplied by π to the πππ. And now you should be able to see the relationship between the two forms and the methods that weβre using.

This time, π§ squared is equal to two root three squared multiplied by π to the power of two multiplied by four π by three π. And, once again, we know that two root three squared is 12. And two multiplied by four π by three is eight π by three. And, once again, changing our argument into the principal argument by subtracting two π, we can see that π§ squared is equal to 12π to the two π by three π.