2In a rectangular system, you go left or right and up or down. How is it Different?In a rectangular system, you go left or right and up or down.In a trigonometric or polar system, you have a direction to travel and a distance to travel in that direction.
3Remember a complex number has a real part and an imaginary part Remember a complex number has a real part and an imaginary part. These are used to plot complex numbers on a complex plane.The absolute value or modulus of z denoted by z is the distance from the origin to the point (a, b).Imaginary AxisThe 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.zbReal Axisamodified for quadrant and so that it is between 0 and 2
4Trigonometric Form of a Complex Number aNote: You may use any other trig functions and their relationships to the right triangle as well as tangent.
5Imaginary Axis Real Axis Plot the complex number and then convert to trigonometric form:Imaginary AxisFind the modulus rr1 ́Real AxisFind the argument but in Quad II
6It 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.2Notice that is the same as plotting
8Multiplying Complex Numbers To multiply complex numbers in rectangular form, you would FOIL and convert i2 into –1.To multiply complex numbers in trig form, you simply multiply the rs and and the thetas.The formulas are scarier than they are.
9Multiply the Moduli and Add the Arguments 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.Must FOIL theseReplace i 2 with -1 and group real terms and then imaginary termsMultiply the Moduli and Add the Argumentsuse sum formula for cosuse sum formula for sin
11Dividing Complex Numbers In rectangular form, you rationalize using the complex conjugate.In trig form, you just divide the rs and subtract the theta.
12(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)
13add the arguments (the i sine term will have same argument) 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.
14subtract the arguments divide the modulisubtract the argumentsIn polar form we want an angle between 0 and 360° so add 360° to the -80°In rectangular coordinates:
16Powers of Complex Numbers This is horrible in rectangular form.It’s much nicer in trig form. You just raise the r to the power and multiply theta by the exponent.The best way to expand one of these is using Pascal’s triangle and binomial expansion.You’d need to use an i-chart to simplify.
17Roots of Complex Numbers There will be as many answers as the index of the root you are looking forSquare root = 2 answersCube root = 3 answers, etc.Answers will be spaced symmetrically around the circleYou divide a full circle by the number of answers to find out how far apart they are
18General Process Problem must be in trig form Take the nth root of n. All answers have the same value for n.Divide theta by n to find the first angle.Divide a full circle by n to find out how much you add to theta to get to each subsequent answer.
19k starts at 0 and goes up to n-1 This is easier than it looks. The formulak starts at 0 and goes up to n-1This is easier than it looks.
20Example 1. Find the 4th root of 81 2. Divide theta by 4 to get the first angle.3. Divide a full circle (360) by 4 to find out how far apart the answers are.List the 4 answers.The only thing that changes is the angle.The number of answers equals the number of roots.