### Video Transcript

Given that π is equal to six root two minus six root two π, write π in trigonometric form.

In this question, weβre given a complex number π written in algebraic form. Thatβs the form π plus ππ, where π and π are real numbers. And we need to convert this complex number into the trigonometric form. So letβs start by recalling what we mean by the trigonometric form of a complex number. This is also known as the polar form. Itβs the form π multiplied by cos of π plus π sin of π, where the value of π is the magnitude of π and π is the argument of π. Therefore, to write π in trigonometric form, we need to find its magnitude and its argument.

Letβs start by finding the magnitude of π. We recall we can find the magnitude of a complex number written in algebraic form by taking the square root of the sum of the squares of its real and imaginary parts. In other words, the magnitude of π plus ππ for real numbers π and π is the square root of π squared plus π squared. Applying this to π, we get the magnitude of π is equal to the square root of six root two squared plus negative six root two squared. And we can then evaluate this expression. Six root two squared is 72. So the magnitude of π is root 72 plus 72, which is the square root of 144, which we can calculate is 12. And this is the value of π in the trigonometric form of π.

Next, we need to determine the argument of π. Weβll do this by sketching π onto an Argand diagram. Remember, to plot π on an Argand diagram, its π₯-coordinate will be the real part of π and the π¦-coordinate will be the imaginary part of π. In this case, its π₯-coordinate will be six root two and its π¦-coordinate will be negative six root two. And since its π₯-coordinate is positive and its π¦-coordinate is negative, π lies in the fourth quadrant of an Argand diagram. We want to use this to determine the argument of π. Remember, the argument of π is the angle the line segment from π to the origin on an Argand diagram makes with the positive real axis measured counterclockwise.

We can determine this angle by constructing the following right triangle. And weβll call the measure of this angle in our right triangle π. Then, we can see that in this right triangle, the side opposite angle π has length six root two and the side adjacent to angle π also has length six root two. Therefore, by using right-triangle trigonometry, tan of π is equal to six root two over six root two. Then, we can evaluate this for π. First, six root two over six root two is equal to one, and we know the inverse tan of one is π by four. So π is equal to π by four. If we then call the argument of π π, we can see that the measure of π and the measure of π add to one full rotation. Itβs two π.

Therefore, the argument of π is two π minus π by four, which we can calculate is seven π by four. And this is the value of π in the trigonometric form of our complex number. And before we substitute these values into our trigonometric form, there is one thing worth pointing out. We could have also measured the argument of π clockwise from the positive real axis. This would have given us negative π by four. This would also give us a valid trigonometric form of π. However, weβll just substitute π is equal to 12 and π is equal to seven π by four into the trigonometric form of our complex number. This then gives us our final answer. π is equal to 12 multiplied by the cos of seven π by four plus π sin of seven π by four.