Video: MATH-ALG+GEO-2018-S1-Q13

If 𝑧 = (1 + √3𝑖)^𝑛 and |𝑧| = 8, find the principal amplitude of the number 𝑧.

04:25

Video Transcript

If 𝑧 equals one plus root three 𝑖 all to the power of 𝑛 and the modulus of 𝑧 is eight, find the principal amplitude of the number 𝑧.

Now 𝑧 is some power of the complex number one plus root three 𝑖. And we know that the modulus of 𝑧 is eight. Putting these together then, we’re given that the modulus of one plus root three 𝑖 to the power of 𝑛 is eight. And what we’re required to find is the principal amplitude of 𝑧, which is one plus root three 𝑖 all to the power of 𝑛. So we’re required to find the amplitude of one plus root three 𝑖 to the power of 𝑛. We’ll worry about making sure it’s the principal amplitude later.

This question involves the modulus and amplitude of the power of a complex number. And so the following facts should come to mind. The modulus of the 𝑛th power of a complex number π‘Ž is the 𝑛th power of the modulus of the complex number π‘Ž. And the amplitude of the 𝑛th power of a complex number π‘Ž is 𝑛 times the amplitude of the complex number π‘Ž. This gives us some hope of a strategy. We can use the given information to find the value of 𝑛 and then use that value of 𝑛 to find the amplitude of one plus root three 𝑖 all to the power of 𝑛. Let’s see if this works.

First, we employ the fact that the modulus of π‘Ž to the 𝑛 is the modulus of π‘Ž all to the power of 𝑛. And so the modulus of one plus root three 𝑖 all to the power of 𝑛 is equal to eight. And we can find the modulus of this complex number one plus root three 𝑖. The modulus of the general complex number π‘₯ plus 𝑦𝑖, where both π‘₯ and 𝑦 are real, is the square root of π‘₯ squared plus 𝑦 squared.

We substitute one for π‘₯ and root three for 𝑦, finding that the modulus of one plus root three 𝑖 is the square root of one squared plus root three squared. One squared is one, and root three squared is three by definition. So we have the square root of one plus three all to the power of 𝑛 equals eight. One plus three is four, and the square root of four is two. So we have two to the power of 𝑛 equals eight. And we know that two cubed is eight. And so 𝑛 must be three.

Let’s clear away our working and use this value of 𝑛 to find the amplitude of 𝑧. This is what we are required to find, the amplitude of one plus root three 𝑖 to the power of 𝑛. And we can use our fact about the amplitude of a power of a complex number, substituting one plus root three 𝑖 for π‘Ž. Now we know the value of 𝑛. It’s three. But what about the value of the amplitude of one plus root three 𝑖?

We can represent one plus root three 𝑖 on an argand diagram. It’s around here. The important thing is that it lies in the first quadrant. The amplitude of this complex number is the angle measured anticlockwise from the positive real axis. We can see that there’s a right-angled triangle hiding in the diagram, where the length of the side opposite the angle πœƒ is the imaginary part of our complex number. That’s root three. And the length of the side adjacent is the real part, one.

We can see then that’s tan πœƒ, which is the opposite of the adjacent, is root three over one, which is root three. And we recognize root three as the value of the tangent of a special angle. Tan 60 degrees is root three. Or in radians tan πœ‹ by three is root three. And so the amplitude πœƒ of one plus root three 𝑖 is πœ‹ by three. We can substitute this value too. So 𝑛 times the amplitude of one plus root three 𝑖 is three times πœ‹ by three, which is πœ‹.

Okay, this is the amplitude of the number 𝑧. But is it the principal amplitude? Recall that the principal amplitude must be greater than negative πœ‹ and less than or equal to πœ‹. And of course πœ‹ falls within this range. So we were lucky. πœ‹ is the principal amplitude of the number 𝑧.

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