Binomial Theorem and Induction with Trigonometry

[Wellsi]

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 doesn't 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

 

[Wellsi]

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)

 

[056593]

Ok... Go searching

 

[HallsofIvy]

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)

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))?

 

[sutupidmath]

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