A Generalized trigonometric identity for Cos(x_1++x_n)?

In summary, the group is discussing the existence of an explicit formula for the sum of products of Sin(x_i) & Cos(x_i). It is understood that there will be 2^(n-1) number of terms in the sum, with each term being a product of n number of Sin(x_i) &/ Cos(x_i). The link provided does not directly answer the question, but it does have related information. The case where all x_i are equal is given by the multiple angle formula. The group is also discussing possible formulas for converting the product of K number of sinx_i and (n-k) number of cosx_i into a sum of Sin and cosine terms. Possible proofs and formulas will be posted soon.
  • #1
NaturePaper
70
0
Hi Everyone,
Do there exist any explicit formula for Cos(x_1+x_2+...+x_n) as a sum of products of Sin(x_i) & Cos(x_i)? Or we need to expand using Cos(A+B), Sin(A+B) again & again?
If it exists then what is about Sin(x_1+x_2+...+x_n)?

[It is understood that there will be 2^(n-1) number of terms in the sum each of which will be
a product of n number of Sin(x_i) &/ Cos(x_i)]

Regards
Naturepaper
 
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  • #2
http://en.wikipedia.org/wiki/Trigonometric_identity

That link has some things that are related, but not exactly the same.

I'm not sure if a formula can be found, but you can try it. Expand it for a couple of cases, try to find a formula and use induction.
 
  • #3
Thank you Gib Z for your reply. But the link you given doesn't answer the question. It was for infinite sum whereas I need for finite terms. Anyway I'm waiting for more reply...Cheers
NaturePaper
 
  • #4
  • #5
NaturePaper, the same page gives
[tex]\sec(\theta_1 + \cdots + \theta_n) = \frac{\sec\theta_1 \cdots \sec\theta_n}{e_0 - e_2 + e_4 - \cdots}.[/tex]
From that, it's obvious that
[tex]\cos(\theta_1 + \cdots + \theta_n) = \frac{e_0 - e_2 + e_4 - \cdots}{\sec\theta_1 \cdots \sec\theta_n}.[/tex]
 
  • #6
@adriank,
oh, very nice-I missed it. Thanks.

Intuitively, Sin(x_1+x_2+...+x_n)=sinx_1.sinx_2...sinx_n(e_1-e_3+e_4-...), where e_i are in terms of cotx_i. I'll check it.

Actually I need a formula to convert the product of K number of sinx_i and (n-k) number of cosx_i into sum of Sin and cosine terms.

If I get the expressions for Sin(x_1+x_2+...+x_n) and cos(x_1+x_2+...+x_n), after some manipulation I can get the required formula.

Thanks & Regards
NaturePaper
 
  • #7
Got it.

Sin(x_1+x_2+...+x_n)=cosx_1cosx_2...cosx_n(e_1-e_3+e_4-...).

By the way, I'll post a possible proof and detailed formula soon.

cheers,
NaturePaper
 

Related to A Generalized trigonometric identity for Cos(x_1++x_n)?

1. What is a generalized trigonometric identity for Cos(x_1++x_n)?

A generalized trigonometric identity for Cos(x_1++x_n) is a formula that represents the relationship between the cosine of the sum of multiple angles (x_1, x_2, ..., x_n) and the product of the cosines of those angles.

2. How is a generalized trigonometric identity for Cos(x_1++x_n) derived?

A generalized trigonometric identity for Cos(x_1++x_n) is derived using the properties of cosine function and the addition formula for cosine.

3. What is the significance of a generalized trigonometric identity for Cos(x_1++x_n)?

Generalized trigonometric identities for Cos(x_1++x_n) are important in mathematics and physics as they allow for simplification of complex trigonometric expressions and are used in various applications such as calculating the values of trigonometric ratios in triangles and solving trigonometric equations.

4. Can a generalized trigonometric identity for Cos(x_1++x_n) be applied to any number of angles?

Yes, a generalized trigonometric identity for Cos(x_1++x_n) can be applied to any number of angles as long as they are all included in the summation.

5. Are there other generalized trigonometric identities?

Yes, there are other generalized trigonometric identities for other trigonometric functions such as sine and tangent. These identities involve the sums or products of multiple angles and can also be derived using the properties and addition formulas of trigonometric functions.

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