Question Video: Finding the Argument of the Power of Complex Numbers Mathematics • Third Year of Secondary School

Video Transcript

Given that 𝑍 is equal to negative 30 plus 30𝑖, determine the principal amplitude of 𝑍 to the fifth power.

In this question, we’re given a complex number 𝑍 written in algebraic form. That’s the form 𝑎 plus 𝑏𝑖, where 𝑎 and 𝑏 are real numbers. We need to use this to determine the principal amplitude of 𝑍 to the fifth power. To answer this question, let’s start by recalling what we mean by the principal amplitude of a complex number. The principal amplitude or principal argument of a complex number is the angle the line segment between 𝑍 and the origin on an Argand diagram makes with the positive real axis, where we restrict this angle to be between negative 𝜋 and 𝜋 in radians and negative 180 and 180 degrees in degrees. In this question, we’re going to work in degrees.

Therefore, to answer this question, we’re first going to need to determine the argument of the complex number 𝑍 to the fifth power. Since we need to determine a complex number to an integer exponent, we’ll do this by recalling de Moivre’s theorem. This tells us for a complex number written in trigonometric form, that’s 𝑟 times cos of 𝜃 plus 𝑖 sin of 𝜃, where 𝑟 is greater than or equal to zero and 𝜃 is any real number, then for any integer value of 𝑛, 𝑟 times cos of 𝜃 plus 𝑖 sin of 𝜃 all raised to the 𝑛th power is equal to 𝑟 to the 𝑛th power multiplied by the cos of 𝑛𝜃 plus 𝑖 sin of 𝑛𝜃. In other words, when we raise a complex number to an integer exponent of 𝑛, we raise its magnitude to that value of 𝑛 and we multiply its argument by 𝑛.

To apply this to find 𝑍 to the fifth power, we’re going to need to write 𝑍 in trigonometric form. And to do this, we need to find the values of 𝑟 and 𝜃 which are the magnitude of 𝑍 and the argument of 𝑍. Let’s start with the magnitude of 𝑍. That’s the distance between 𝑍 and the origin on an Argand diagram. We can find this by finding the square root of the sums of the squares of the real and imaginary parts of 𝑍. That’s the square root of negative 30 squared plus 30 squared, which, if we evaluate, is equal to 30 root two.

However, it is worth pointing out we don’t actually need to find this value. The magnitude of 𝑍 tells us the distance between 𝑍 and the origin in an Argand diagram. And we can see the magnitude of the complex number 𝑍 does not affect its argument when raised to an integer exponent. Therefore, the magnitude of 𝑍 will not affect the argument of 𝑍 to the fifth power. And in particular, this means it won’t affect its principal amplitude either. However, it can be useful to see how to write 𝑍 in trigonometric form anyway.

Next, we need to determine the argument of 𝑍. And we’ll do this by first working out which quadrant in an Argand diagram 𝑍 lies in. Since the real part of 𝑍 is negative 30 and its imaginary part is 30, its 𝑥-coordinate will be negative 30 and its 𝑦-coordinate will be 30. This means it lies in the second quadrant. We can then determine the argument of 𝑍 by recalling the following result. If 𝑎 plus 𝑏𝑖 is a complex number written in algebraic form in the second quadrant of an Argand diagram, then the argument of 𝑎 plus 𝑏𝑖 is equal to the inverse tan of 𝑏 divided by 𝑎 plus 180 degrees. This allows us to determine the argument of 𝑍. Our value of 𝑏, the imaginary part of 𝑍, is 30. And our value of 𝑎, the real part of 𝑍, is negative 30. So, the argument of 𝑍 is the inverse tan of 30 divided by negative 30 plus 180 degrees.

We can then evaluate this. The inverse tan of negative one is negative 45 degrees, giving us the argument of 𝑍 is 135 degrees. We can then use this to write 𝑍 in trigonometric form. 𝑍 is 30 root two multiplied by the cos of 135 degrees plus 𝑖 sin of 135 degrees. Now, we can use de Moivre’s theorem to raise both sides of the equation to the fifth power. Since five is an integer exponent, when we raise 𝑍 to the fifth power, we raise its magnitude to the fifth power and we multiply its argument by five. 𝑍 to the fifth power is 30 root two raised to the fifth power multiplied by the cos of five times 135 degrees plus 𝑖 sin of five times 135 degrees.

Now, we’re only interested in finding the principal amplitude of 𝑍 to the fifth power. We can do this directly from its argument. First, we’ll simplify this argument. Five multiplied by 135 degrees is 675 degrees. And now the principal argument or principal amplitude of this value is the equivalent angle between negative 180 and 180 degrees, including 180 degrees.

We can then determine the principal amplitude of 𝑍 to the fifth power by recalling both cosine and sine are periodic with a period of 360 degrees. In other words, we can add and subtract integer multiples of 360 degrees from its argument. Therefore, if we subtract 360 degrees from 675 degrees, we don’t change the value. This then gives us 315 degrees. However, this is not in the given interval. So we subtract another 360 degrees to get negative 45 degrees, which is in this interval.

Therefore, we were able to show if 𝑍 is negative 30 plus 30𝑖, then the principal amplitude of 𝑍 to the fifth power is negative 45 degrees.