What is the argument of the complex number 𝑎 plus 𝑏𝑖, where 𝑎 is greater than zero and 𝑏 is greater than zero?
In this question, we’re asked to find the arguments of the complex number 𝑎 plus 𝑏𝑖. We’re told that 𝑎 is positive and 𝑏 is positive. And whenever we’re asked to find the arguments of a complex number, it’s always a good idea to plot this onto an Argand diagram. And remember, in 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, we want to plot the complex number 𝑎 plus 𝑏𝑖, which we’ll call 𝑍, onto our Argand diagram. And we’re told that the values of 𝑎 and 𝑏 are positive. So this is an example of the algebraic form of a complex number. And when a complex number is given in algebraic form, it’s easy to read off the imaginary and real parts for our complex number. The real part of 𝑍 will be the constant on its own, which in this case is 𝑎. And the imaginary part of 𝑍 will be the coefficient of 𝑖, which in this case is 𝑏. So in our case, on our Argand diagram, the coordinates of the point 𝑍 should be 𝑎, 𝑏. And in the question, we’re told that both of these are positive. So we can put these onto our Argand diagram. And this means we can plot the point 𝑍.
Now, we’re ready to start finding the argument of our complex number 𝑍. We recall that the argument of a complex number 𝑍 is the angle that the line segment from the origin to 𝑍 on an Argand diagram makes with the positive real axis. So in this case, we can mark this angle on our Argand diagram as 𝜃. And it’s worth pointing out if we measure this angle counterclockwise, we give it as a positive value. And if we measure it clockwise, we give it as a negative value. So in our case, we can see that 𝜃 is going to be positive because we’ve measured it counterclockwise. And in fact, we can also see that 𝜃 is an acute angle.
Now, to find our angle 𝜃, we’re going to construct the following right-angled triangle. We just go vertically down from 𝑍 to the real axis and then across to the origin. The height of this right-angled triangle is going to be the vertical coordinate of point 𝑍, which is the imaginary part of 𝑍, which we know is 𝑏. Similarly, the base of this right-angled triangle is going to be the real part of 𝑍, which we know is 𝑎. So now our angle 𝜃 is an angle in the right-angled triangle, where we know the length of the opposite side to 𝜃 and we know the length of the adjacent side to 𝜃.
And we know, by using trigonometry, the tan of 𝜃 will be equal to the length of its opposite side divided by the length of its adjacent side. So for our triangle, we have the tan of 𝜃 is equal to 𝑏 divided by 𝑎.
We want to find the value of 𝜃 because this is the argument of our complex number. So to do this, we’re going to want to take the inverse tangent of both sides of the equation. And before we do this, there is one thing worth pointing out. Because our value of 𝑏 and 𝑎 are both positive, their quotient is going to be positive. And at this point, we’ve not discussed whether we’re taking our angle 𝜃 in radians or degrees. It doesn’t matter. We’ll get the same answer regardless. However, we’ll just use degrees in this case.
We know if 𝑐 is positive, then the inverse tan of 𝑐 will be bigger than zero degrees and less than 90 degrees. So when we take the inverse tangent of both sides of this equation, we get 𝜃 is equal to the inverse tan of 𝑏 divided by 𝑎. And we know that this value is between zero degrees and 90 degrees. And this was all just to check that we got the correct answer that 𝜃 was the positive acute angle in this triangle.
So this result gives us a useful formula for calculating the arguments of a complex number 𝑎 plus 𝑏𝑖, where 𝑎 and 𝑏 are positive. And another way of saying this is the point 𝑍 lies in the first quadrant of our Argand diagram. We showed that the argument of this complex number is the inverse tan of 𝑏 divided by 𝑎.