Compute , and a_{2} if sin x is approximated using a_{0} a_{1} sin x = a_{0}*Q_{0} + a_{1}*Q_{1} + a_{2}*Q_{2} where Q_{1} are the Legendre polynomials of degree i

To solve the problem, we need to approximate \(\sin x\) using a series expansion in terms of the Legendre polynomials \(Q_i(x)\), where \(Q_i(x)\) represents the Legendre polynomial of degree \(i\). The approximation is given by:

\[
\sin x \approx a_0 Q_0(x) + a_1 Q_1(x) + a_2 Q_2(x)
\]

### Step 1: Identify the Legendre Polynomials

The first few Legendre polynomials are:

- \(Q_0(x) = 1\)
- \(Q_1(x) = x\)
- \(Q_2(x) = \frac{1}{2}(3x^2 - 1)\)

### Step 2: Express \(\sin x\) in terms of these polynomials

We want to find the coefficients \(a_0\), \(a_1\), and \(a_2\) such that:

\[
\sin x \approx a_0 \cdot 1 + a_1 \cdot x + a_2 \cdot \frac{1}{2}(3x^2 - 1)
\]

### Step 3: Determine the coefficients \(a_0\), \(a_1\), and \(a_2\)

The coefficients can be found using the orthogonality property of Legendre polynomials:

\[
a_i = \frac{\int_{-1}^{1} \sin(x) Q_i(x) \, dx}{\int_{-1}^{1} \left[Q_i(x)\right]^2 \, dx}
\]

We now compute these integrals:

1. **Coefficient \(a_0\)**:
    \[
    a_0 = \frac{\int_{-1}^{1} \sin(x) \cdot 1 \, dx}{\int_{-1}^{1} 1 \cdot 1 \, dx}
    \]

2. **Coefficient \(a_1\)**:
    \[
    a_1 = \frac{\int_{-1}^{1} \sin(x) \cdot x \, dx}{\int_{-1}^{1} x^2 \, dx}
    \]

3. **Coefficient \(a_2\)**:
    \[
    a_2 = \frac{\int_{-1}^{1} \sin(x) \cdot \frac{1}{2}(3x^2 - 1) \, dx}{\int_{-1}^{1} \left[\frac{1}{2}(3x^2 - 1)\right]^2 \, dx}
    \]

### Step 4: Simplify the integrals

Now, let's evaluate these integrals:

1. **\(a_0\) Integral:**

    The integral of \(\sin(x)\) over \([-1, 1]\) is:

    \[
    a_0 = \frac{\int_{-1}^{1} \sin(x) \, dx}{2}
    \]

    \[
    a_0 = \frac{-\cos(x)\Big|_{-1}^{1}}{2} = \frac{-\cos(1) + \cos(-1)}{2} = 0
    \]

2. **\(a_1\) Integral:**

    Since \(\sin(x)\) is an odd function and \(x\) is odd, the integral of their product over the symmetric interval \([-1, 1]\) is:

    \[
    a_1 = 0
    \]

3. **\(a_2\) Integral:**

    Let's evaluate this integral:

    \[
    a_2 = \frac{1}{2} \cdot \frac{\int_{-1}^{1} \sin(x) \cdot (3x^2 - 1) \, dx}{\int_{-1}^{1} \left[\frac{1}{2}(3x^2 - 1)\right]^2 \, dx}
    \]

    The integral can be tricky and might require numerical methods or advanced techniques for an exact value, but generally, \(a_2\) will not be zero and would give a nontrivial contribution to the approximation.

### Final Answer:

- \(a_0 = 0\)
- \(a_1 = 0\)
- \(a_2\) can be computed (typically requiring numerical integration)

Would you like a more detailed computation of \(a_2\) or have any other questions?

Here are some related questions:

1. How do Legendre polynomials relate to the Fourier series?
2. What is the orthogonality property of Legendre polynomials?
3. How can you use the Gram-Schmidt process to derive Legendre polynomials?
4. How does the approximation improve with higher degree polynomials?
5. What are some applications of Legendre polynomials in physics?

**Tip:** Legendre polynomials are widely used in solving differential equations in physics, particularly in spherical harmonics.

Learn how to approximate sin x using Legendre polynomials with detailed steps and computations. Understand the orthogonality property and series approximation in mathematics.

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