This is a free lesson from our course in Trigonometry This lesson will help you to understand the concept of complex numbers in polar form and how to write complex numbers in terms of a distance from the origin and a direction (or angle) from the positive horizontal axis. It is explained by the instructor with the help of video, how you can find the real (horizontal) and imaginary (vertical) components in terms of r (the length of the vector) and (the angle made with the real axis).Let's take case of x + iy. Using the relationships x = r cos , and y = r sin and r = (x2 + y2), the values of r and can be obtained and then replace x and y in z = x + iy, which results in polar form of complex number      z = x + iy = r(cos + i sin ) (More text below video...)
Other useful lessons:
 The Complex Number System Multiplication of Complex Numbers
(Continued from above) where r is the absolute value (or modulus) and is the argument of the complex number and is equal to tan-1(y/x).
Polar form is sometimes called the trigonometric form as well. The polar form of a complex number is especially useful when you're working with powers and roots of a complex number. For example, to write the complex number (rectangular form) z = 1 + i into polar form, first you compare it with z = x + yi and see that z = 1 + i has rectangular coordinates (1, 1). Now find the values of r and and then plug in r cos for x and r sin for y in z = x + yi and factor out r. It will yield to 2(cos /4 + i sin /4).
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