Help with Proof: Sum of Cosines Up to n Terms

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    Proof
In summary, the expression \sum_{k=0}^{k=n}(nCk * cos(kx)) = cos(nx/2)*(2cos(x/2))^n can be simplified to (1+e^{ix})^n = 2^n cos^n (x/2) cos (nx/2) by substituting z=exp(ix) and taking the real part.
  • #1
shen07
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\(\displaystyle \sum_{k=0}^{k=n}(nCk * cos(kx)) = cos(nx/2)*(2cos(x/2))^n\)
 
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  • #2
I also know that i have to use

\(\displaystyle
z=exp(ix)
\)
\(\displaystyle
(1+z)^n = 2^n cos^n (x/2) cos (nx/2) \)
 
  • #3
Got the answer..
 
  • #4
shen07 said:
Got the answer..

Hi shen07 (Wave),

Welcome to MHB! Sorry we couldn't help you quickly enough this time. I'm sure that in the future you'll find guidance with something you are stuck on. Care to share your answer so others may see it?

Jameson
 
  • #5
The answer could be obtained by choosing the real part after substituting $z=e^{ix}$

Hence we have

\(\displaystyle \Large (1+e^{ix})^n= e^{\frac{ixn}{2}}\left(e^{\frac{-ix}{2}}+e^{\frac{ix}{2}}\right)^n= 2^n e^{\frac{ixn}{2}}\cos^n \left(\frac{x}{2}\right)\)

Clearly the answer is the real part of the above expression .
 

FAQ: Help with Proof: Sum of Cosines Up to n Terms

What is the formula for finding the sum of cosines up to n terms?

The formula for finding the sum of cosines up to n terms is sum = cos(x) + cos(2x) + cos(3x) + ... + cos(nx), where x is the angle in radians and n is the number of terms.

How do I know when to stop adding terms in the sum of cosines formula?

You can stop adding terms when you have reached the desired number of terms, n. Another way to determine when to stop is when the value of the last term added is very small (close to 0), indicating that adding more terms would not significantly change the overall sum.

Can the sum of cosines formula be used for any angle measurement?

Yes, the sum of cosines formula can be used for any angle measurement in radians. However, if the angle is given in degrees, it must be converted to radians before using the formula.

How can the sum of cosines formula be derived?

The formula for the sum of cosines can be derived using the trigonometric identity cos(a+b) = cos(a)cos(b) - sin(a)sin(b). By substituting different values for a and b in this identity and simplifying, the formula for the sum of cosines can be obtained.

What is the significance of the sum of cosines formula in mathematics?

The sum of cosines formula is important in many areas of mathematics, including calculus, trigonometry, and complex analysis. It is used to solve various problems involving periodic functions, such as finding the average value of a function over a given interval.

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