Complex Number Yr 13 De Moivre Etc Flashcards
Why do all complex roots of unity add to make 0?
This is because they form a gerometric sequence that sums to 0
De moivre theroem
Z n = cos n theta + is in n theta = (cos theta + isin theta) ^n
If there is a minus sign in the de moivre, how ti tackle this so can still use
Must use idea cos even function and sin odd, make the angle - and now use
What if it says sin + icos
How ti adjust
Take I out of the equation, remembering I x -I =1
Now you have case before where need odd and even functions
How to show all roots of unity sun to 0
Well first root is 1, omega is root with smallest angle, and these roots increase to omega ti n-1
This forms geometric sequence, sum to n gives yiu a formula
But because w to n is = 1, it adds to firm ZERO
What are properties of roots of unity
All roots will have modulus 1
The roots will be of constsnt argument away from each other = 2 pi /n
All form polygon
How to express sin ti the 6 in other forms of sin ti make it easier
Just use identify Z ti n + Z ti - n or other snd rearrange for sin, then lower it to 6, grouo again, simply
How to express cos 4 theta in terms of cos theta
Well this is the REAL part of the expansion of Z to the power of 4
So do thst snd grouo again
Use same identify e^ix + e^-x
Same as Z^n + z^-n = 2costheta
2isintheta similalry
Ei theta =
Cos theta + isin theta
Remember when raising to power of n make sure to do basic numbers too
What’s the complex conjugate of e To power i theta ?
E ^ x i theta
REMEMBEE WHEN PLOTTING IN ARGAND DISGRAM WHAT TI DO
INDICATE THE ANGLES
For the retarded questions about midpoint how ti solve?
BEST WAY?
Probably an easy way but all I can think of is find the mid point in numbers, now find the mod and argument and manipulate
Remember that to make it even, square and write square root if that etv
LIT TRY FIND THE MID AND AGRYMENT IF THE POINT
- ARG IS EASY, USE ANGLES
- MOD IS EASY BEVAUSE YIU KNIW MOD IF ACTUAL OOINTS, BECOMES A RUGHT SNGLED TRAINGLE AND USE TRIG
NKW ITS SOO CLEAN
