Standard to polar form uses two equations - Pythagorean Theorem and the inverse tangent function.
A complex number has two elements, the real and imaginary components. The real component is the 'x' value and the imaginary component is the 'y' value. In the same manner as vectors, you can convert the x and y components to their matching r and (theta) components.
For example, use the complex number #3 - 4i#.
The real component is 3 and the imaginary component is -4.
The radius (r) is another name for the hypotenuse of a right triangle formed with these two numbers, so an application of the Pythagorean Theorem gives #sqrt(3^2+(-4)^2# and the radius is equal to 5.
For the angle ( #theta# ), the inverse tangent function uses the ratio of y/x, or in the case of complex numbers, imaginary/real.
One final step - determine the quadrant that the complex number is located within. If the real (x) is positive, and the imaginary (y) is negative, that means quadrant IV.
If your calculator is like mine, it gives an answer of about -53.1. But if we are in quadrant IV, that means we are 53.1 degrees shy of a full 360 degree circle. The actual angle then is about 306.9 degrees. The absolute most effective way to figure out how to get the right angle is to make a quick sketch of the original complex number on an x-y graph. That way, you can immediately see which quadrant you are in.
Quadrant I - the calculator gives the right answer
Quadrant II and III - add 180 to the answer the calculator gives
Quadrant IV - add 360 to the answer the calculator gives
But regardless of these shortcuts, nothing beats making a quick sketch so you can visualize what you are working with.
Therefore, we would express the polar form of the answer as: