- #1
xuying1209
- 4
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if n is an odd, cosπ/n+cos3π/n+cos5π/n+...+cos(2n-1)π/n is equal to what?
And how can I prove it??
And how can I prove it??
Odd Cosn Problem: What is cosπ/n+cos3π/n+...+cos(2n-1)π/n? Prove It!
- Thread starter xuying1209
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In summary, the sum of cosines of n terms, where n is an odd number, is equal to 0 if n is greater than 1. This is because when adding the real parts of the 2nth roots of unity, they are symmetric about the imaginary axis, resulting in a sum of 0. For n=1, the sum is equal to -1. The notation used was [itex] cos(i\pi/n)[/tex] for i= 1 to n-1.
- #2
HallsofIvy
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For n= 1, cos(pi/1)= -1.
For n> 1, n odd, essentially you are adding the real parts of the 2nth roots of unity. Since those roots are symmetric about the imaginary axis, the sum is 0.
For n> 1, n odd, essentially you are adding the real parts of the 2nth roots of unity. Since those roots are symmetric about the imaginary axis, the sum is 0.
- #3
tehno
- 375
- 0
Ahh that it was... almost racked my brains out 'cause "π" I read as n ( not [itex]\pi[/itex]) ...
- #4
robert Ihnot
- 1,059
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I am not sure what is being asked. Is this [tex]\sum cos(n_i)/n_i, or \sum cos(pi*n_i/n_i), or what?[/tex]
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- #5
HallsofIvy
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I interpreted as sum of [itex] cos(i\pi/n)[/tex] for i= 1 to n-1.
FAQ: Odd Cosn Problem: What is cosπ/n+cos3π/n+...+cos(2n-1)π/n? Prove It!
What is the Odd Cosn Problem?
The Odd Cosn Problem is a mathematical problem that involves finding the value of the expression cosπ/n+cos3π/n+...+cos(2n-1)π/n. It is also known as the "Odd Cosine Sum Problem" or the "Odd Cosine Number Problem".
What is the significance of this problem?
The Odd Cosn Problem has significant applications in various fields, including signal processing, quantum computing, and mathematical physics. It is also a challenging mathematical problem that has puzzled many mathematicians for decades.
How can the Odd Cosn Problem be solved?
The Odd Cosn Problem can be solved using complex analysis techniques, specifically the Euler's formula and the geometric series formula. It involves converting the cosine terms into complex exponentials and then using summation formulas to evaluate the expression.
Is there a general solution for the Odd Cosn Problem?
Yes, there is a general solution for the Odd Cosn Problem. It is given by the formula cosπ/n+cos3π/n+...+cos(2n-1)π/n = 1/2 * cot(π/2n) * sin(nπ). This formula can be derived using complex analysis and can be easily verified using mathematical induction.
Can you provide a proof for the solution of the Odd Cosn Problem?
Yes, the solution for the Odd Cosn Problem can be proved using mathematical induction and complex analysis techniques. The proof involves manipulating and simplifying the expression using Euler's formula, geometric series formula, and trigonometric identities. It can be a bit complex but is a well-established proof in mathematics.