Using the same function f (x) = sin (x) over the interval 0 ≤ x ≤ π, construct an
even function of period π, sketch the function over three periods, and obtain an appropriate
sine-cosine Fourier series for this function

We are tasked with constructing an even function from \( f(x) = \sin(x) \) over the interval \( 0 \leq x \leq \pi \), extending it to have a period of \( \pi \), and then finding the sine-cosine Fourier series for this new function.

### Step 1: Construct an Even Function

An even function satisfies \( f(-x) = f(x) \). The original function \( f(x) = \sin(x) \) is not even since \( \sin(-x) = -\sin(x) \). However, we can construct an even function from \( \sin(x) \) by defining a new function \( g(x) \) such that:
\[
g(x) = 
\begin{cases} 
\sin(x) & \text{if } 0 \leq x \leq \pi, \\
\sin(-x) = -\sin(x) & \text{if } -\pi \leq x \leq 0. \\
\end{cases}
\]
Thus, for \( g(x) \), we define:
\[
g(x) = \sin(|x|), \quad \text{for all } x.
\]

This function is even because:
\[
g(-x) = \sin(|-x|) = \sin(|x|) = g(x).
\]

### Step 2: Extend \( g(x) \) to a Periodic Function

We now extend \( g(x) \) to be periodic with period \( \pi \). This means that the function repeats every \( \pi \) units. The extended function can be written as:
\[
g(x + \pi) = g(x) \quad \text{for all } x.
\]

Thus, we construct a periodic function that is even and satisfies:
\[
g(x) = \sin(x) \quad \text{for } 0 \leq x \leq \pi, \quad \text{and repeats every } \pi.
\]

### Step 3: Sketch the Function

The function \( g(x) \) over three periods (from \( -3\pi \) to \( 3\pi \)) looks like a series of "bumps" due to the periodic nature of the sine function being reflected for negative \( x \). At each integer multiple of \( \pi \), the function reaches zero, and between these points, it oscillates in a smooth, continuous manner. 

### Step 4: Fourier Series for the Even Periodic Function

Since \( g(x) \) is even, the sine terms in its Fourier series vanish, and we only have cosine terms. The general form of a Fourier cosine series for an even periodic function with period \( \pi \) is:
\[
g(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos\left(\frac{2n\pi x}{\pi}\right).
\]

To find the coefficients \( a_0 \) and \( a_n \), we use the following formulas:
\[
a_0 = \frac{2}{\pi} \int_0^\pi \sin(x) \, dx,
\]
\[
a_n = \frac{2}{\pi} \int_0^\pi \sin(x) \cos\left(\frac{2n\pi x}{\pi}\right) \, dx \quad \text{for } n \geq 1.
\]

#### Calculate \( a_0 \):
\[
a_0 = \frac{2}{\pi} \int_0^\pi \sin(x) \, dx = \frac{2}{\pi} \left[-\cos(x)\right]_0^\pi = \frac{2}{\pi} [1 - (-1)] = \frac{4}{\pi}.
\]

#### Calculate \( a_n \) for \( n \geq 1 \):
Using integration by parts or a standard Fourier series table, we know that:
\[
a_n = 0 \quad \text{for all } n \geq 1.
\]

Thus, the Fourier series reduces to just the constant term:
\[
g(x) = \frac{2}{\pi}.
\]

### Conclusion

The even, periodic function constructed from \( f(x) = \sin(x) \) is a constant function over each interval due to the cancellation of higher-order terms. The Fourier series for this function is simply:
\[
g(x) = \frac{2}{\pi}.
\]

Let me know if you'd like further details on any of the steps.

### Related Questions:
1. What is the general formula for the Fourier series of an odd function?
2. How do you find the Fourier sine series for \( f(x) = \sin(x) \) without symmetry extensions?
3. What is the significance of even and odd functions in Fourier series?
4. How do you compute Fourier series for non-periodic functions?
5. How do the boundary conditions of \( f(x) \) affect the Fourier coefficients?

### Tip:
When constructing an even function from an odd function, reflect the odd part symmetrically and ensure the function maintains periodicity by adjusting its intervals accordingly.

Using the same function f(x) = sin(x) over the interval 0 ≤ x ≤ π, construct an even function of period π, sketch the function over three periods, and obtain an appropriate sine-cosine Fourier series for this function.

Learn how to construct an even function from sin(x) over the interval 0 ≤ x ≤ π, extend it to a periodic function of period π, and obtain its sine-cosine Fourier series. This tutorial covers key topics in Fourier analysis, including Fourier coefficients, periodic functions, and symmetry considerations.

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