8. The expansion of the periodic function f(x) in a sine-cosine Fourier series is:

f(x)= 1 2 + 2 pi ( (sin 2x)/2 + (sin 3x)/3 +***)

b. f(x)= 8 pi ( (sin x)/1 + (sin 3x)/3 +***)

f(x)= 1 2 + 2 pi ( (cinx)/1 + coxx 1 ++***)

f(x)= 1 2 + 2 pi ( (cor*pi*x)/1 - (cos 3pi*x)/3 +***)

It looks like you're trying to express different variations of a Fourier series for a periodic function \( f(x) \), but the notation is unclear in some parts. Let’s break down what seems to be intended in each of these options and correct them if necessary:

### 1. Fourier Series General Form

For a periodic function \( f(x) \), the general form of a Fourier sine-cosine expansion is:

\[
f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin(nx) \right)
\]

where:
- \( a_0 \) is the constant term.
- \( a_n \) and \( b_n \) are the Fourier coefficients for the cosine and sine terms respectively.

### Interpretation of Your Options:

#### (a) \( f(x) = \frac{1}{2} + 2\pi \left( \frac{\sin(2x)}{2} + \frac{\sin(3x)}{3} + \cdots \right) \)

This resembles the sine expansion of a periodic function. In this case, the function only contains sine terms with coefficients of the form \( \frac{2\pi}{n} \). However, the first sine term should have \( \sin(x) \), not \( \sin(2x) \), as part of the typical Fourier sine series.

A corrected form could be:

\[
f(x) = \frac{1}{2} + 2\pi \left( \frac{\sin(x)}{1} + \frac{\sin(2x)}{2} + \frac{\sin(3x)}{3} + \cdots \right)
\]

#### (b) \( f(x) = 8\pi \left( \frac{\sin(x)}{1} + \frac{\sin(3x)}{3} + \cdots \right) \)

This is another sine series but lacks the constant term, and the coefficient outside the sum is \( 8\pi \). It implies a stronger scaling of the sine terms. The form could represent a specific periodic function where only sine terms are present, but without the constant term \( \frac{a_0}{2} \), it may not be the full Fourier series.

#### (c) \( f(x) = \frac{1}{2} + 2\pi \left( \frac{\sin(x)}{1} + \cos(x) + \cdots \right) \)

Here, the notation seems off (e.g., "cinx" and "coxx"). This could be meant to represent both sine and cosine terms in the series, but it’s not properly formatted. A more appropriate expansion could be:

\[
f(x) = \frac{1}{2} + 2\pi \left( \frac{\sin(x)}{1} + \frac{\cos(x)}{1} + \cdots \right)
\]

#### (d) \( f(x) = \frac{1}{2} + 2\pi \left( \frac{\cos(\pi x)}{1} - \frac{\cos(3\pi x)}{3} + \cdots \right) \)

This appears to be a cosine-only expansion, where terms like \( \cos(\pi x) \) and \( \cos(3\pi x) \) appear. This could correspond to an even function that only contains cosine terms. The general structure seems correct for a cosine series.

### Summary

From the four options, each represents different types of Fourier series expansions, but the notation needs some adjustment:

1. **(a)** looks like a sine-only series but needs the first term \( \sin(x) \) instead of \( \sin(2x) \).
2. **(b)** is another sine series but scaled by \( 8\pi \) without a constant term.
3. **(c)** is likely intended to mix sine and cosine terms, but the notation is unclear.
4. **(d)** is a cosine-only series, potentially for an even function.

Would you like details on how to fully derive these Fourier series or have any specific questions on Fourier analysis?

### Relative Questions:
1. How are Fourier coefficients determined for a given periodic function?
2. What is the difference between a sine-only and cosine-only Fourier series?
3. How does the periodicity of the function affect its Fourier series expansion?
4. Can a Fourier series have only sine or cosine terms, and under what conditions?
5. How does scaling affect the terms of a Fourier series expansion?

### Tip:
To construct a Fourier series, always ensure the function is periodic, and properly compute both sine and cosine coefficients based on the symmetry properties of the function.

The expansion of the periodic function f(x) in a sine-cosine Fourier series is:

f(x)= 1 2 + 2 pi ( (sin 2x)/2 + (sin 3x)/3 +***)

b. f(x)= 8 pi ( (sin x)/1 + (sin 3x)/3 +***)

f(x)= 1 2 + 2 pi ( (cinx)/1 + coxx 1 ++***)

f(x)= 1 2 + 2 pi ( (cor*pi*x)/1 - (cos 3pi*x)/3 +***)

Understand the Fourier series expansion for periodic functions. This problem involves sine-cosine expansions, covering key concepts like periodic functions, sine and cosine terms, and their coefficients. Suitable for university-level students.

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