Complex nos (1 Viewer)

[mstellatos]

C = 1 + cos x + cos 2x +..... + cos (n-1)x
S= sinx+sin2x+sin3x....+sin(n-1)x.

i) Multiply series S by i ...... did this Ok
ii) Write down C+iS ...did this Ok
iii) If z=cosx+isinx, use de moivres to express c+is as a series in terms of z. .....did this Ok
iv) Hence show that C + iS = (1-z^n)/ (1-z) .......did this part OK.
Next part I need help with this

Using the following results
sin A -sin B =2cos(A+B)/2 sin(A-B)/2
cos A - cosB = -2 sin (A+B)/2sin(A-B)/2
show that C= [sin 1/2nx cos 1/2 (n-1)x] /(sin1/2x) and
S = [ sin 1/2nx sin 1/2(n-1)x] / (sin1/2x)

 

[Affinity]

you are basically finding the imaginary and real parts of (1-z^n)/ (1-z)


(1-z^n)/ (1-z)
=[1 - cos(nx) - isin(nx)]/[1 - cos(x) - isin(x)]
=[2sin^2(nx/2) - 2i*sin(nx/2)cos(nx/2)]/[2sin^2(x/2) - 2i*sin(x/2)cos(x/2)]
=[sin(nx/2)/sin(x/2)][(sin(nx/2) - i cos(nx/2)/(sin(x/2) - icos(x/2))]
=[sin(nx/2)/sin(x/2)][(cos(pi/2 - nx/2) - isin(pi/2 - nx/2)/(cos(pi/2 - x/2) - isin(pi/2 - x/2))]
=[sin(nx/2)/sin(x/2)][(cos(nx/2 - pi/2) + isin(nx/2 - pi/2)/(cos(x/2 - pi/2) - isin(x/2 - pi/2))]
=[sin(nx/2)/sin(x/2)][cos((nx/2 - pi/2) - (x/2 - pi/2)) + isin((nx/2 - pi/2) - (x/2 - pi/2))]
=[sin(nx/2)/sin(x/2)][cos((n-1)x/2) + isin((n-1)x/2)]

and the result follows immediately

 

[mstellatos]

Thank you but what does the asterisk mean in line 3 and I don't understand line 3 where you have introduced the p what does the sin(pi/2-nx/2)-isin(pi/2-nx/2) come from...and then in the line that follows cos(nx/2-pi/2)+isin(nx/2-pi/2) come from\
THanks sorrry to be dumb

 

[Affinity]

pi = "pie" and remember that sin(x) = cos(pi/2 - x)
* just means multiply

 

[mstellatos]

thanks but when you have cos(pi/2-nx/2)- isin(pi/2-nx/2)..why can you change it in the next line to cos (nx/2-pi/2) + isin (nx/2-pi/2)

and the next line when you are subracting the denominator is that de moivre's rule .
thanks again for your help

 

[STx]

wheres this Q from?

 

[Affinity]

cos(x) = cos(-x)
sin(x) = -sin(-x)

 

[Blind Ref]

Great work team!!

Now, I'll be honest....

I'm doin 4U, comin 2nd in d class and I have absolutely no idea wot u have just done...?

Maybe that's cos it looks funny typed up!!

Oh well...its only 4U maths!!

 

[Affinity]

^ means exponentiate, everything else should be self explanatory. try copying the contents to a piece of paper with the usual notation

 

[Riviet]

Also note that a / means to divide (as in a fraction). Pay attention to brackets too as they will determine what's inside what.