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Binomial Theorem and Induction with Trigonometry

  1. Dec 5, 2006 #1
    Ok i need help to calculate the co-efficients of certain terms in the binomial expansion for example:

    (3 + (5/X)^2)^10
    what is the coefficient of x^8?

    I hope that question works sorry if it doesnt i did just make it up then......
    if you know of any like it please help!

    also, an excerise left me confused as anything the other day - it was integrating mathematical induction and trigonometric relationships.... ill post the questions in about an hour does anyone know how these work the entire K+1 thing throws me off
     
  2. jcsd
  3. Dec 5, 2006 #2
    ok here are the questions i was going to post:
    5) Use the principle of mathematical induction to prove that:
    sinx + sin3x + sin5x + .... + sin(2n-1)x = (1-cos2nx)/(2sinx)
    for all positive intergers n and hence find the value of
    sin(pi/7) + sin(3pi/7) + sin(5pi/7) .... + sin(13pi/7)

    6)Use the principle of mathematical induction to prove that:
    cosx x cos2x x cos4x x cos8x ...... x cos((2^n)x)= (sin(2^n)x)/(2^n x sinx)
     
  4. Mar 20, 2007 #3
    Ok... Go searching
     
  5. Mar 20, 2007 #4

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    First, please do not use "x" both as a variable and to mean multiplication. Use parentheses or "*" instead.

    Now, what have you tried? What happens in each of those when n= 1?
    How do you go from sinx + sin3x + sin5x + .... + sin(2n-1)x to sinx + sin3x + sin5x + .... + sin(2(n+1)-1)x ?

    How do you go from (cosx)(cos2x)(cos4x)(cos8)...... (cos((2^n)x)) to (cosx)(cos2x)(cos4x)(cos8)...... (cos((2^(n+1))x))?
     
  6. Mar 20, 2007 #5
    LIke HallsofIvy instructed, first you have to prove that for n=1 that equation holds true. So prove that sinx=(1-cos2nx)/2sinx , for n=1. After you prove this, than suppose that the equation also holds true for n,( or n=k,it is the same) so you suppose that the equation:
    sinx+ sin3x+....+sin(2n-1)x=(1-cos2nx)/2sinx is true, or is valid, this is called the inductin hypothesis(hi)
    and after this you have to prove that the above equation also holds true for n+1(or n=k+1),

    so what you have to prove is this:

    sinx+sin3x+.....+sin(2n-1)x+sin(2(n+1)-1)=(1-cos2(n+1)x)/2sinx

    The other two problems follow almost the same pattern.
    Now do you kno what to do?

    anyone, correct me if i am wrong
    i hope it helps
     
    Last edited: Mar 20, 2007
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