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

Video Transcript

Given that 𝑍 is equal to nine plus three 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 complex number 𝑍. We need to give our answer rounded to the nearest two decimal places. To do this, we’re going to first need to recall what we mean by the argument of a complex number 𝑍 and what it means for this argument to be principal. The argument of a complex number 𝑍 is the angle the ray from the origin to 𝑍 makes with the positive real axis on an Argand diagram. What this means is whenever we’re asked to find the argument of a complex number 𝑍, it’s a good idea to sketch this onto an Argand diagram.

In this question, we’re asked to find the argument of 𝑍 is equal to nine plus three 𝑖, so we’ll put this onto our Argand diagram. Remember, on an Argand diagram, the horizontal axis represents the real part of our complex number and the vertical axis represents the imaginary part of our complex number. In our case, 𝑍 is given in algebraic form. That’s the form 𝑎 plus 𝑏𝑖 where 𝑎 and 𝑏 are real numbers. And this means we can just read off the real and imaginary parts of our complex number 𝑍. The real part is going to be the constant on its own. That’s nine. And the imaginary part is going to be the coefficient of 𝑖. That’s three.

This means on our Argand diagram, 𝑍 will have horizontal coordinate nine and vertical coordinate three. So we can plot this onto our Argand diagram. We’re now ready to try and find the argument of 𝑍. To do this, we need to draw a ray from the origin to our point 𝑍. The argument of 𝑍 is any angle that this ray makes with the positive real axis. For example, we can mark that angle 𝜃, and because 𝜃 is measured counterclockwise, 𝜃 will be positive, but this is not the only possible angle. We could also measure this angle clockwise, so there’s a lot of different possible options for the argument of 𝑍, which is why we have something called the principal argument of 𝑍.

If 𝜃 is any argument of the complex number 𝑍, then we say that 𝜃 is the principal argument of 𝑍 if one of the following two things are true. If we’re measuring our angle in radians, we must have that 𝜃 is bigger than negative 𝜋 and 𝜃 is less than or equal to 𝜋. However, if we’re measuring our angle in degrees, then we must have that 𝜃 is bigger than negative 180 degrees and less than or equal to 180 degrees. In this question, we’re going to measure our angle in degrees. However, usually, we measure these angles in radians, and we can see we’ve already marked the principal argument of 𝑍 onto our diagram because we know this value of 𝜃 is positive and we also know it’s going to be an acute angle.

We have a few different methods of finding our angle of 𝜃. The easiest way is to construct the following right-angle triangle. We just go vertically down from 𝑍 to our real axis and then across to the origin. Then the base of this right-angle triangle is going to be the modulus of the real part of 𝑍, which is nine, and the height of this right-angle triangle is going to be the modulus of the imaginary part of 𝑍. It’s going to be equal to three. We can then find our angle of 𝜃 by using trigonometry. The tan of an angle 𝜃 in a right-angle triangle will be equal to the length of the side opposite 𝜃 divided by the length of the side adjacent to 𝜃. In our case, we get the tan of 𝜃 is equal to three divided by nine. And we can solve for our value of 𝜃 by taking the inverse tangent of both sides of this equation.

Now, we could simplify three over nine to be one-third. However, it’s not necessary. We get 𝜃 is the inverse tan of three over nine. And there is something worth pointing out here. If we were solving this as an equation, we would know there are multiple solutions, so we always need to check our diagram to make sure we’re finding the correct value. In this case, 𝜃 is an acute angle, and it’s positive because it’s measured counterclockwise. And if we evaluate this with our calculator set to degrees mode, we get 𝜃 is equal to 18.434 and this continues degrees, which is a positive acute value. In fact, it’s the only one of these which solves this equation. So this is the correct value of angle 𝜃.

And it’s worth pointing out here there were other ways we could’ve found this angle. But usually, the easiest way is just to find an acute angle on our diagram and use this to find the principal argument. So we’re almost done. There’s only one more thing the question wants us to do. We need to give our answer to two decimal places. And to do this, we look at the third decimal place of our answer which is four, which is less than five. So we know we need to round down, which is our final answer. Therefore, given the complex number 𝑍 is equal to nine plus three 𝑖, we were able to find the principal argument of 𝑍 to two decimal places. We got that it was equal to 18.43 degrees.