Binomial Theorem and Induction with Trigonometry

In summary, the conversation involves a request for help with calculating coefficients in a binomial expansion and understanding mathematical induction and trigonometric relationships. Two specific questions are mentioned, one involving a proof using mathematical induction and the other involving a product of cosine functions. The participants in the conversation provide guidance and ask for clarification on steps taken by the individual seeking help.
  • #1
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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
 
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  • #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)
 
  • #3
Ok... Go searching
 
  • #4
Wellsi said:
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))?
 
  • #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:

1. What is the binomial theorem?

The binomial theorem is a mathematical formula used to expand expressions that contain binomials, which are expressions with two terms. It states that the coefficients of each term in the expansion can be found by using combinations and the exponents of the terms decrease in a specific pattern.

2. How is the binomial theorem used in trigonometry?

In trigonometry, the binomial theorem can be used to expand expressions involving trigonometric functions, such as sine and cosine. This can be helpful in simplifying and solving trigonometric equations.

3. What is mathematical induction and how is it related to the binomial theorem?

Mathematical induction is a proof technique used to prove mathematical statements that depend on a variable. It involves showing that a statement is true for a base case and then proving that if the statement is true for any given case, it is also true for the next case. The binomial theorem can be proven using mathematical induction.

4. Can the binomial theorem be used with any type of binomial?

Yes, the binomial theorem can be used with any type of binomial, as long as it follows the general form of (a + b)^n, where a and b are any real numbers and n is a positive integer.

5. What are the applications of the binomial theorem in real life?

The binomial theorem has various applications in fields such as physics, engineering, and finance. It can be used to model and analyze natural phenomena, such as population growth and radioactive decay. It can also be applied in calculating probabilities and making predictions in financial markets.

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