 The complex number x plus iy can be identified with the point with rectangular coordinates x, y. But the rectangular coordinates of the point x, y can be expressed in terms of the polar coordinates r, theta, where the polar and rectangular coordinates are related by the following equations. Consequently, we can express the complex number x plus iy in polar or trigonometric form as r cosine theta plus i sine theta, where r is the modulus and theta is the argument. We often use the shorthand r sis theta. So for example, we might try to find the trigonometric form of 1 minus i. So the complex number 1 minus i corresponds to the point 1 negative 1. Now we want to convert the coordinates from rectangular form into polar form. So remember our fundamental relationship. And so the complex number 1 negative i corresponds to the point 1 negative 1. And so we find r for theta. We note that so theta is either negative pi fourths or 3 pi fourths. Now note that 1 negative 1 is in the fourth quadrant. So we have to choose theta to be negative pi fourths. And so this gives us our trigonometric form with modulus square root 2 and argument negative pi fourths. As another example, let's try to rewrite in trigonometric form 1 plus square root 3 i. So again the number in rectangular form corresponds to a pointed rectangular coordinates and converting our pointed rectangular coordinates into polar coordinates gives us r equals and tan of theta should be y divided by x. So we have tan theta equals square root 3. And since our point 1 root 3 is in the first quadrant, one solution is going to be pi thirds. And so our point in rectangular coordinates gives us our point in trigonometric coordinates with modulus 2 and argument pi thirds. It's important to keep in mind that the polar coordinates for a point are not unique. Several polar coordinates can describe the same point. And this will become important so let's find three different trigonometric forms for 8. So we note that the value 8 is the same as 8 plus 0 i and in geometric form this corresponds to the point 8 0. And we can find three different sets of polar coordinates for the point 8 0. First if we don't rotate at all and just go from the pole straight to the point 8 distance of 8 which will give us polar coordinates 8 0. We could also do a full rotation and then go out distance 8 and that gives us polar coordinates 8 to pi and we can't even go around twice and then go out distance 8 and that gives us a third form 8 for pi.