Sin (x)as Legendre series with argument cosx

To express \(\sin(x)\) as a Legendre series in terms of the argument \(\cos(x)\), we expand \(\sin(x)\) in a series of Legendre polynomials \(P_n(\cos(x))\). Legendre polynomials form an orthogonal basis for functions on the interval \([-1, 1]\), and \(\cos(x)\) takes values in this interval.

### General Expansion of a Function in Terms of Legendre Polynomials
Any function \(f(x)\) can be expanded as:

\[
f(x) = \sum_{n=0}^{\infty} a_n P_n(\cos(x)),
\]
where the coefficients \(a_n\) are given by:

\[
a_n = \frac{2n + 1}{2} \int_{-1}^{1} f(\cos(x)) P_n(\cos(x)) \, d(\cos(x)).
\]

Now, let's compute the coefficients \(a_n\) for \(f(x) = \sin(x)\).

### 1. Change of Variable
We need to express \(\sin(x)\) in terms of \(\cos(x)\). We set \(t = \cos(x)\), so that \(dt = -\sin(x)dx\). Thus, the integral becomes:

\[
a_n = \frac{2n + 1}{2} \int_{-1}^{1} \sin(\cos^{-1}(t)) P_n(t) \, dt.
\]

### 2. Simplification Using Trigonometric Identity
Using the identity \(\sin(\cos^{-1}(t)) = \sqrt{1 - t^2}\), the expansion becomes:

\[
a_n = \frac{2n + 1}{2} \int_{-1}^{1} \sqrt{1 - t^2} P_n(t) \, dt.
\]

### 3. Coefficient Calculation
The coefficients \(a_n\) can now be evaluated using standard methods for Legendre series expansion involving \(\sqrt{1 - t^2}\). These are non-zero only for odd values of \(n\), as \(\sin(x)\) is an odd function.

Thus, \(\sin(x)\) can be expanded as a Legendre series involving only odd Legendre polynomials.

### Legendre Series for \(\sin(x)\)
The final form of the Legendre series for \(\sin(x)\) is:

\[
\sin(x) = \sum_{n=1, 3, 5, \dots} a_n P_n(\cos(x)).
\]

This is the Legendre series expansion of \(\sin(x)\) in terms of \(\cos(x)\). The coefficients \(a_n\) would involve specific integrals that can be computed for each \(n\), but generally, this series is convergent for all \(x\).

Would you like further details on calculating these coefficients or another specific part of the process?

### Five Related Questions:
1. How do we derive the Legendre polynomials?
2. What are the orthogonality properties of Legendre polynomials?
3. Can you expand other trigonometric functions like \(\cos(x)\) in a Legendre series?
4. How does the convergence of the Legendre series depend on the function being expanded?
5. How do we apply Legendre series expansions in solving physical problems?

**Tip:** The orthogonality of Legendre polynomials simplifies the calculation of expansion coefficients significantly.

Sin (x) as Legendre series with argument cosx

Explore how to express sin(x) as a Legendre series in terms of cos(x). This problem involves orthogonal expansions using Legendre polynomials, detailed formulas for expansion coefficients, and trigonometric identities. Suitable for undergraduate students in mathematics or physics.

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