Use the Regula falsi method to solve the equation x = cosx, 

The **Regula Falsi method** (also known as the False Position method) is a root-finding algorithm that combines aspects of the **Bisection Method** and **Secant Method**. We use this method to solve the equation \( x = \cos(x) \) by first rewriting it in the form:

\[
f(x) = x - \cos(x) = 0
\]

### Steps of the Regula Falsi Method:
1. **Initial guesses**: Choose two points \( x_0 \) and \( x_1 \) such that \( f(x_0) \cdot f(x_1) < 0 \). This ensures there is a root between \( x_0 \) and \( x_1 \).

2. **Calculate the root approximation**: The next approximation \( x_2 \) is given by the formula:

\[
x_2 = x_1 - \frac{f(x_1)(x_1 - x_0)}{f(x_1) - f(x_0)}
\]

3. **Update**: Replace either \( x_0 \) or \( x_1 \) based on the sign of \( f(x_2) \).

4. **Repeat**: Continue until the root is found within a desired tolerance.

### Solving \( x = \cos(x) \):

1. **Define the function**:
   \[
   f(x) = x - \cos(x)
   \]

2. **Initial guesses**:
   Let’s choose \( x_0 = 0 \) and \( x_1 = \frac{\pi}{2} \) (approx 1.57). Notice that:

   \[
   f(0) = 0 - \cos(0) = 0 - 1 = -1
   \]
   \[
   f\left(\frac{\pi}{2}\right) = \frac{\pi}{2} - \cos\left(\frac{\pi}{2}\right) = 1.57 - 0 = 1.57
   \]

   So, \( f(0) \cdot f(1.57) < 0 \), meaning there is a root between 0 and \( \frac{\pi}{2} \).

3. **Calculate the next approximation**:
   Using the Regula Falsi formula:
   
   \[
   x_2 = x_1 - \frac{f(x_1)(x_1 - x_0)}{f(x_1) - f(x_0)}
   \]
   Substituting values \( x_0 = 0 \), \( x_1 = 1.57 \), \( f(0) = -1 \), and \( f(1.57) = 1.57 \):

   \[
   x_2 = 1.57 - \frac{(1.57)(1.57 - 0)}{1.57 - (-1)} = 1.57 - \frac{(1.57)(1.57)}{2.57} = 1.57 - 0.959 = 0.611
   \]

4. **Update and repeat**:
   Check the sign of \( f(0.611) \) and update the interval accordingly. Continue iterating until the result converges to a desired tolerance.

### Conclusion:
Repeating this process several times, we eventually converge to the solution. The approximate root of \( x = \cos(x) \) is \( x \approx 0.7391 \).

Would you like more details or have any questions?

### Related Questions:
1. How does the Regula Falsi method compare to other root-finding methods like the Bisection Method?
2. How can the accuracy of the Regula Falsi method be improved?
3. What is the geometric interpretation of the Regula Falsi method?
4. Why is \( x = \cos(x) \) a transcendental equation, and what does that imply about its solutions?
5. What is the stopping criterion in the Regula Falsi method?

### Tip:
In the Regula Falsi method, convergence might be slower than in other methods like Newton's method, but it doesn't require the derivative of the function, making it useful when derivatives are difficult to compute.

Use the Regula falsi method to solve the equation x = cos(x).

Learn how to apply the Regula Falsi method to solve the transcendental equation x = cos(x). Follow a detailed step-by-step guide, using initial guesses and iterative calculations to approximate the solution to x ≈ 0.7391. This method leverages concepts from numerical analysis and root-finding algorithms.

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