What is the Sum of a Finite Sine Series?

In summary, a finite sine series is a mathematical expression that combines a finite number of sine functions with their coefficients to model periodic phenomena in science and engineering. The sum of a finite sine series is calculated by adding up the individual sine functions with their coefficients using mathematical techniques or a calculator. It is used in scientific research to analyze and predict the behavior of periodic phenomena. However, it can only accurately represent functions with a finite number of harmonics, unlike an infinite sine series which has an infinite number of terms and can accurately represent any periodic function.
  • #1
shehry1
44
0
Can anyone tell me what is the sum of a finite series of Sines.

[tex]\sum_{n=1}^N \sin^2 (n \phi) [/tex]. I am going through a text and it gives it as (N+1)/2. I tried to derive it. The N comes out ok when you use the half angle identity but I can't figure out a general rule for the Cosines that appear
 
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  • #2
What is [itex]\phi[/itex]?
 
  • #3
HallsofIvy said:
What is [itex]\phi[/itex]?

[tex] \phi = \pi s /(N+1) [/tex] where s = 1,2...N
 
  • #4
resolve sin^2 into 1-cos something. Put it in terms of complex exponentials. you'll get a geometric series.
 

FAQ: What is the Sum of a Finite Sine Series?

What is a finite sine series?

A finite sine series is a mathematical expression that consists of a finite number of sine functions, each with its own coefficient, added together. It is used to model periodic phenomena in various fields of science and engineering.

How is the sum of a finite sine series calculated?

The sum of a finite sine series is calculated by adding up the individual sine functions with their respective coefficients. This can be done using mathematical techniques such as the trigonometric identities or by using a calculator or computer program.

What is the purpose of using a finite sine series in scientific research?

A finite sine series is used in scientific research to model and analyze periodic phenomena such as sound waves, light waves, and electrical signals. It allows scientists to understand and predict the behavior of these phenomena and make accurate calculations and predictions.

Can a finite sine series accurately represent any periodic function?

No, a finite sine series can only accurately represent periodic functions that have a finite number of harmonics. If a periodic function has an infinite number of harmonics, then an infinite sine series would be needed to accurately represent it.

How is a finite sine series different from an infinite sine series?

A finite sine series has a limited number of terms, whereas an infinite sine series has an infinite number of terms. This means that a finite sine series can only approximate a periodic function, while an infinite sine series can accurately represent it.

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