Question Video: Finding the Principal Argument of a Complex Number Mathematics

Video Transcript

Given that 𝑍 is negative three minus seven 𝑖, find the principal argument of 𝑍 rounded to the nearest two decimal places.

In this question, we’re given a complex number 𝑍. And we’re asked to find the principal argument of our value of 𝑍. And we need to give our answer to two decimal places. To answer this question, we first need to recall what we mean by an argument of 𝑍 and what it means for an argument to be the principal argument of 𝑍. We recall that we say the argument of a complex number 𝑍 is equal to 𝜃 when 𝜃 is the angle 𝑍 makes with the positive real axis on an Argand diagram.

And there’s a few things worth pointing out about this definition. First, when we say the angle 𝑍 makes with the positive real axis, we actually mean the angle between the positive real axis and the ray connecting the origin to our point 𝑍. However, it can be easier to think about this as the angle 𝑍 makes with the positive real axis.

Next, since 𝑍 will represent the angle measured from the positive real axis, this means there’s going to be multiple values for our argument. For example, an angle of zero, an angle of 360, and an angle of 720 would all represent the same value. And to get around this problem, we introduced the concept of a principal argument. For a principal argument measured in radians, we demand that 𝜃 would be bigger than negative 𝜋 and less than or equal to 𝜋. And the same is true if we want to measure this angle in degrees. We demand that 𝜃 will be bigger than negative 180 degrees and less than or equal to 180 degrees.

And it’s worth pointing out here when we say a positive angle, we mean this angle is measured counterclockwise. And when we say a negative angle, we mean that this angle is measured clockwise. Now that we have this information, we’re ready to find the argument of our complex number 𝑍 given to us in the question. And since we need to find an angle, the easiest way to do this will be to sketch our point on an Argand diagram.

So, we start by sketching our axes. Remember, in an Argand diagram, the horizontal axes represents the real part of our complex number and the vertical axis represents the imaginary part of our complex number. We want to plot 𝑍 is equal to negative three minus seven 𝑖 onto our Argand diagram. And we can see that 𝑍 is given in algebraic form. That’s the form 𝑎 plus 𝑏𝑖, where 𝑎 and 𝑏 are real numbers. So we can use this to find the imaginary and real parts of our value of 𝑍.

The real part of 𝑍 is our value of 𝑎, the constant on its own, which in this case is negative three. And the imaginary part of 𝑍 is the value of 𝑏 or the coefficient of 𝑖, which, in this case, is negative seven. So, the horizontal coordinate of 𝑍 on our Argand diagram is its real part, negative three, and its vertical coordinate is its imaginary part, negative seven. So, we can plot 𝑍 onto our Argand diagram.

Next, we’ll sketch the ray connecting 𝑍 to the origin. And finally, we can sketch the argument of 𝑍 onto our diagram. And since we want to find the principal argument of 𝑍, so we’re only allowed to do half a turn at most, we’re going to need to go clockwise from the positive real axis. We’ve shown the argument of 𝑍 is going to be negative.

And now, there’s a lot of different ways of finding the exact value of 𝜃. Most of these are going to involve some kind of trigonometry. It doesn’t matter which method you would prefer. In this video, we’re going to find the value of 𝛼, shown on our diagram. To find our value of 𝛼, we’re going to need to use trigonometry. First, let’s find the sides of our triangle. We know that the height is seven and the width is three. This is just the absolute value of each of the coordinates of point 𝑍, and we also know this is a right-angled triangle.

So, 𝛼 is an angle in a right-angled triangle where we know the length of the opposite side and the length of the adjacent side. So, by using trigonometry, tan of 𝛼 is equal to seven over three. In other words, 𝛼 is the inverse tan of seven over three. And by using our calculator set in degrees mode, we get that 𝛼 is equal to 66.80, and this continues, degrees. And it’s worth pointing out here our value of the angle 𝛼 is positive because we’re calculating the measure of this angle. We can use this to find the measure of angle 𝜃 and then use that to find 𝜃.

In our diagram, we can see that 𝛼 and 𝜃 are two angles on a straight line. The measure of these two angles will add to be 180 degrees. There’s a few different ways of writing this. We’ll just write this as 𝛼 plus the absolute value of 𝜃 is 180 degrees. And we can just use this to find the value of 𝜃. We’ll subtract 𝛼 from both sides of this equation. This gives us the measure of angle 𝜃 is 180 degrees minus 𝛼 which, to three decimal places, is 113.198 degrees. And we know from our diagram since 𝜃 is measured clockwise, we want it to be negative. So, we’ve shown, to three decimal places, 𝜃 is negative 113.198 degrees.

However, the question wants us to give this to two decimal places. So, we check our third decimal place to see if we need to round up or round down. This is eight, which is bigger than five. So, we need to round up. And because the next digit is nine, we need to carry this over. So, to two decimal places, our angle of 𝜃 is negative 113.20 degrees, which is our final answer.

Therefore, we were able to find the principal argument of 𝑍 is equal to negative three minus seven 𝑖 to two decimal places. We got that this angle was negative 113.20 degrees.