The Complex Plane; DeMoivre's Theorem- converting to trigonometric form

The magnitude or modulus of z is the same as r. We can take complex numbers given as and convert them to polar form. Recall the conversions: Real Axis Imaginary Axis Do Now: Plot the complex number: factor r out

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. The magnitude or modulus of z denoted by z is the distance from the origin to the point (x, y). y x z  The angle formed from the real axis and a line from the origin to (x, y) is called the argument of z, with requirement that 0   < 2 . modified for quadrant and so that it is between 0 and 2  Use these conversions to find the trigonometric form of the do now

The magnitude or modulus of z is the same as r. We can take complex numbers given as and convert them to polar form. Recall the conversions: Real Axis Imaginary Axis y x z = r  Plot the complex number: Find the polar form of this number. 1  factor r out Shorthand:

The angle is between -  and  Real Axis Imaginary Axis y x z = r  1  but in Quad II

It is easy to convert from polar 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 polar form, you would plot the angle and radius. 2 Notice that is the same as plotting

Convert the following:

Do Now: Convert the following to trigonometric form:

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 polar 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 r values and A dd the A ngles

(This says to multiply two complex numbers in polar form, multiply the r values and add the angles) (This says to divide two complex numbers in polar form, divide the r values and subtract the angles)

multiply the r valuesadd the angles (the i sine term will have same angle) If you want the answer in complex coordinates simply compute the trig functions and multiply the 24 through.

multiply the r valuesadd the angles (the i sine term will have same angle) If you want the answer in complex coordinates simply compute the trig functions and multiply the 24 through.

divide the r valuessubtract the angles In polar form we want an angle between 0 ° and 180° “PRINCIPAL ARGUMENT”

divide the r valuessubtract the angles In polar form we want an angle between 0 ° and 180° PRINCIPAL ARGUMENT In rectangular coordinates:

Today’s theorem is used to raise complex numbers to powers. Do Now: simplify (foil)

Today’s theorem is used to raise complex numbers to powers. Do Now: simplify

Abraham de Moivre ( ) DeMoivre’s Theorem You can repeat this process raising complex numbers to powers. Abraham DeMoivre did this and proved the following theorem: This says to raise a complex number to a power, raise the r value to that power and multiply the angle by that power.

This theorem is used to raise complex numbers to powers. Instead let's convert to polar form and use DeMoivre's Theorem: CONVERT: APPLY DEMOIVRE’S THEOREM: CONVERT BACK:

This theorem is used to raise complex numbers to powers. It would be a lot of work to find you would need to FOIL and multiply all of these together and simplify powers of i --- UGH! Instead let's convert to polar form and use DeMoivre's Theorem.

This theorem is used to raise complex numbers to powers. It would be a lot of work to find you would need to FOIL and multiply all of these together and simplify powers of i --- UGH! Instead let's convert to polar form and use DeMoivre's Theorem. but in Quad II

This theorem is used to raise complex numbers to powers. It would be a lot of work to find

Example: Remember: Convert Apply Demoivre Convert back 1 st : (Put into calculator and see what the value is)

Example: Convert:

Example: Convert: Apply:

Convert back:

Solve the following over the set of complex numbers : We get 1 but we know that there are 3 roots. So we want the complex cube roots of 1. Using DeMoivre's Theorem, we can develop a method for finding complex roots. This leads to the following formula:

N is the root and k goes from 0 to 1 less than the root. We find the number of roots necessary

Let's try this on our problem. We want the cube roots of 1. We want cube root so our n = 3. Can you convert 1 to polar form? (hint: 1 = 1 + 0i) We want cube root so use 3 numbers here Once we build the formula, we use it first with k = 0 and get one root, then with k = 1 to get the second root and finally with k = 2 for last root.

We found the cube roots of 1 were: Let's plot these on the complex plane about 0.9 Notice each of the complex roots has the same magnitude (1). Also the three points are evenly spaced on a circle. This will always be true of complex roots. each line is 1/2 unit

Find the cube roots of: Find z in trig form:

Find the cube roots of:

the cube roots in trig form: Root 1 Root 2 Root 3

the cube roots in complex form:

Find all real and complex cube roots of -8 Z= i Find trig form first:

Find all real and complex roots of -8 Z= i Find trig form first: Now find the 3 roots in trig form

Find all real and complex roots of -8 Z= i the 3 roots in trigonometric form:

Find all real and complex roots of -8 Z= i the 3 roots in complex form:

Do Now: Find all real roots of

Find all real and complex fourth roots of 16 Z= 16+ 0i Find trig form first: 16cis0 Write it 4 times:

Find all real and complex fourth roots of 16 Z= 16+ 0i

Find all real and complex fourth roots of 16 Z= 16+ 0i

Find all real and complex 6 th roots of Z= 1- i

Find all real and complex 6 th roots of Z= 1- i

Find all real and complex 6 th roots of 1- i