Trigonometric Form of Complex Numbers

Real Axis Imaginary Axis Remember a complex number has a real part and an imaginary part. These are used to plot complex numbers on a complex plane. b a z  The angle formed from the real axis and a line from the origin to (a, b) is called the argument of z, with requirement that 0   < 2 . modified for quadrant and so that it is between 0 and 2  The absolute value or modulus of z denoted by z is the distance from the origin to the point (a, b).

Trigonometric Form of a Complex Number  b a  Note: You may use any other trig functions and their relationships to the right triangle as well as tangent. r

Real Axis Imaginary Axis r  ́ Plot the complex number and then convert to trigonometric form: Find the modulus r 1  but in Quad II Find the argument 

It is easy to convert from trigonometric to rectangular form because you just work the trig functions and distribute the r through. If asked to plot the point and it is in trigonometric form, you would plot the angle and radius. 2 Notice that is the same as plotting

use sum formula for sinuse sum formula for cos Replace i 2 with -1 and group real terms and then imaginary terms Must FOIL these Let's try multiplying two complex numbers in trigonometric form together. Look at where we started and where we ended up and see if you can make a statement as to what happens to the r 's and the  's when you multiply two complex numbers. M ultiply the M oduli and A dd the A rguments

(This says to multiply two complex numbers in polar form, multiply the moduli and add the arguments) (This says to divide two complex numbers in polar form, divide the moduli and subtract the arguments)

multiply the moduliadd the arguments (the i sine term will have same argument) If you want the answer in rectangular coordinates simply compute the trig functions and multiply the 24 through.

divide the modulisubtract the arguments In polar form we want an angle between 0 and 360° so add 360° to the -80° In rectangular coordinates: