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Trigonometric Form of a Complex Number

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Complex Numbers Recall that a complex number has a real component and an imaginary component. z = a + bi Argand Diagram Real axis Imaginary axis z = 3 – 2i a bi The absolute value of a complex number is its distance from the origin. The names and letters are changing, but this sure looks familiar.

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The Trig form of a Complex Number

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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.

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Converting from Rectangular form to Trig form Convert z = 4 + 3i to trig form. 1. Find r2. Find3. Fill in the blanks

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Converting from Trig Form to Rectangular Form This ones easy. 1.Evaluate the sin and cos. 2.Distribute in r Convert 4(cos 30 + i sin 30) to rectangular form. 1. Evaluate the sin and cos 2. Distribute the 4.

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Multiplying Complex Numbers To multiply complex numbers in rectangular form, you would FOIL and convert i 2 into –1. To multiply complex numbers in trig form, you simply multiply the rs and add the thetas. The formulas are scarier than it really is.

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Example Rectangular formTrig form

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Dividing Complex Numbers In rectangular form, you rationalize using the complex conjugate. In trig form, you just divide the rs and subtract the theta.

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Example Rectangular form Trig form

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De Moivres Theorem If is a complex number And n is a positive integer Then

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Who was De Moivre? A brilliant French mathematician who was persecuted in France because of his religious beliefs. De Moivre moved to England where he tutored mathematics privately and became friends with Sir Issac Newton. De Moivre made a breakthrough in the field of probability (writing the Doctrine of Chance), but more importantly he moved trigonometry into the field of analysis through complex numbers with De Moivres theorem.

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But, can we prove DeMoivres Theorem? Lets look at some Powers of z.

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Lets look at some more Powers of z.

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It appears that: Assume n=1, then the statement is true. We can continue in the previous manor up to some arbitrary k Let n = k, so that: Now find Proof:

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Eulers Formula We can also use Eulers formula to prove DeMoivres Theorem.

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So what is the use? Find an identity for using Mr. De Moivres fantastic theory Remember the binomial expansion: Apply it: Cancel out the imaginery numbers:

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Now try these:

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Powers of Complex Numbers This is horrible in rectangular form. The best way to expand one of these is using Pascals triangle and binomial expansion. Youd need to use an i-chart to simplify. Its much nicer in trig form. You just raise the r to the power and multiply theta by the exponent.

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Roots of Complex Numbers There will be as many answers as the index of the root you are looking for Square root = 2 answers Cube root = 3 answers, etc. Answers will be spaced symmetrically around the circle You divide a full circle by the number of answers to find out how far apart they are

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The formula k starts at 0 and goes up to n-1 This is easier than it looks. Using DeMoivres Theorem we get

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General Process 1. Problem must be in trig form 2. Take the n th root of r. All answers have the same value for r. 3. Divide theta by n to find the first angle. 4. Divide a full circle by n to find out how much you add to theta to get to each subsequent answer.

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Example 1. Find the 4 th 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. 4.List the 4 answers. The only thing that changes is the angle. The number of answers equals the number of roots.

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