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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.
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Weβre currently given a complex number written in trigonometric form.
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And weβre looking to find π§ squared in exponential form.
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There are two ways we can go about this.
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We can evaluate π§ squared in trigonometric form, and then convert it to exponential form.
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Or we can convert it to exponential form first, and then work out the value of π§ squared.
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Letβs consider both of these methods.
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And to square this complex number, we recall De Moivreβs theorem.
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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.
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And in this example, π is a natural number.
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We can see that the modulus of our complex number π§ is two root three.
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And π, its argument is 240 degrees.
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In fact, at some point, weβre going to have to convert this to radians.
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So, we might as well do that now and get it out of the way.
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To do this, we recall the fact that two π radians is equal to 360 degrees.
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And we can find the value of one degree by dividing through by 360.
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One degree is equal to two π over 360 radians.
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And two π by 360 simplifies to π by 180.
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So, one degree is equal to π over 180 radians.
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So, we can change 240 degrees into radians by multiplying it by π over 180.
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That gives us four π by three.
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And so, we can work out the modulus of π§ squared by squaring the modulus of π§.
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Thatβs two root three squared.
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Root three squared is three.
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So, two root three squared is two squared multiplied by three, which is 12.
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And then, to work out the argument of π§ squared, we multiply the argument of π§ by the power thatβs two.
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Four π by three multiplied by two is eight π by three.
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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.
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And remember, to change a complex number in trigonometric form into exponential form, itβs ππ to the ππ.
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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.
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Remember though, we usually want to represent this using the principal argument.
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Thatβs greater than negative π and less than or equal to π.
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In fact, eight π by three is greater than π.
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So, to find the principal argument, we add or subtract multiples of two π.
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Here, letβs subtract two π from eight π by three.
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Two π is equal to six π over three.
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And when we subtract six π over three from eight π over three, weβre left with two π over three.
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So, in exponential form, π§ squared is 12π to the two π by three π.
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Now, letβs consider the alternative method.
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And that was to convert this complex number into exponential form first and then square it.
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Once again, weβll use this rule.
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A complex number with a modulus of π and an argument π can be represented in exponential form as ππ to the ππ.
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We already solved 240 degrees is equal to four π by three radians.
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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.
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And this time, to find π§ squared, we consider the alternative form of De Moivreβs theorem.
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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.
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And now you should be able to see the relationship between the two forms and the methods that weβre using.
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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.
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And, once again, we know that two root three squared is 12.
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And two multiplied by four π by three is eight π by three.
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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.