given a complex number in the complex plane: -16+12i, write it in polar form

Question

Help, please and thank you!

Answer 1

Okay, so hopefully this makes some sense. There are 4 quadrants on a graph; the top right, then top left, then bottom left, and last bottom right. In other words they are numbered counterclockwise starting at the top right.

From what I remember, when looking at values that have real and imaginary components, the real numbers are measured on the x-axis or horizontal axis, while the imaginary numbers are measured on the y-axis or vertical axis.

In this case, your real number is -16, so that means left of the origin by 16 units. The imaginary number is +12i, so that means 12 units up from the origin. You are in the 2nd quadrant (top left).

Polar coordinates have 2 components, the radius and the angle. The radius is the distance from the origin and the angle is measured counterclockwise from the positive x-axis.

You need pythagorean theorem to get the radius and trigonometry to get the angle.

In this case the radius is (-16)²+12²=r² so r=20.

Now, picture the coordinate -16,12 on a graph, it is on the top left. Now picture the radius going from the origin, up and to the left by a length of 20. Now draw a horizontal line in the right direction to the vertical axis. You should see a right triangle, where the hypotenuse is the radius with length 20. Then the horizontal line is one leg of the triangle with length 16, and the vertical axis down to the origin is the other leg of the triangle with length 12. Since tan is opposite/adjacent, do tan^-1(16/12) which is 53.13 degrees. And since you are in the 2nd quadrant, add 90 degrees to 53.13, which is equal to 143.13 degrees which is the angle you are looking for.

In short, the radius is 20 and the angle is 143.13 degrees counterclockwise from the positive x-axis. I hope this helps.