(2(3x-1.25)sin^3(πx))/3+((9((sin2πx/(2πx)-x))/10)×(sinπx+sin(πx(1.25-2x))-0.0443π(3x-1.25))＝0

To solve the given equation:

\[
\frac{2(3x - 1.25) \sin^3(\pi x)}{3} + \left(\frac{9 \left(\frac{\sin(2\pi x)}{2\pi x} - x \right)}{10} \times \left(\sin(\pi x) + \sin(\pi x (1.25 - 2x)) - 0.0443 \pi (3x - 1.25)\right)\right) = 0
\]

we need to find the value of \(x\) that satisfies this equation. Let's denote this equation as:

\[
f(x) = \frac{2(3x - 1.25) \sin^3(\pi x)}{3} + \left(\frac{9 \left(\frac{\sin(2\pi x)}{2\pi x} - x \right)}{10} \times \left(\sin(\pi x) + \sin(\pi x (1.25 - 2x)) - 0.0443 \pi (3x - 1.25)\right)\right) = 0
\]

### Steps to Solve:

1. **Isolate terms involving \(\sin(\pi x)\)**: Analyze and simplify each term separately.
2. **Use numerical methods**: Given the complexity of the trigonometric terms, numerical methods or graphical methods may be necessary to find the roots.

### Numerical Solution Approach:

Given the complexity, it's often practical to use numerical methods such as the Newton-Raphson method or other root-finding algorithms. Here, we'll sketch a general approach using a numerical solver.

First, let's define the function \( f(x) \) and use numerical methods to find its root.

### Python Code for Numerical Solution:
Here's a Python code snippet using the Newton-Raphson method for root finding:

```python
import numpy as np
from scipy.optimize import fsolve

def f(x):
    term1 = (2 * (3 * x - 1.25) * np.sin(np.pi * x)**3) / 3
    term2 = (9 * (np.sin(2 * np.pi * x) / (2 * np.pi * x) - x)) / 10
    term3 = np.sin(np.pi * x) + np.sin(np.pi * x * (1.25 - 2 * x)) - 0.0443 * np.pi * (3 * x - 1.25)
    return term1 + term2 * term3

# Initial guess for x
initial_guess = 0.5
root = fsolve(f, initial_guess)

print("The root is:", root)
```

Running this code will provide a numerical solution for the value of \( x \).

### Analytical Approach:

To attempt an analytical solution, we need to simplify the terms as much as possible. However, due to the non-linear trigonometric terms, the analytical approach may be impractical without specific values or approximations.

Would you like more details on the numerical solution or any specific step in the analytical approach?

Here are some follow-up questions you might ask:
1. How does the Newton-Raphson method work for solving equations?
2. Can you explain the steps of the fsolve function in Python?
3. What are some common numerical methods for solving non-linear equations?
4. How do you verify the solution of a complex equation like this?
5. What are the advantages and limitations of using numerical methods?

**Tip:** When dealing with complex trigonometric equations, numerical methods can provide practical solutions, especially when analytical solutions are challenging to obtain.

Explore how to solve a complex trigonometric equation involving sine functions and numerical methods. This problem requires understanding trigonometry, numerical root-finding techniques like the Newton-Raphson method, and their applications. Suitable for advanced high school and college students.

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