Approximate root of the function f(x)=cosx-x using method of false position

The **method of false position** (also called the **regula falsi method**) is an iterative technique used to approximate the roots of a function. It works similarly to the bisection method but uses a secant line (linear interpolation) to approximate the root more efficiently.

To find the approximate root of \( f(x) = \cos(x) - x \) using the false position method, follow these steps:

### Step-by-Step Solution:

1. **Define the function**:  
   \[
   f(x) = \cos(x) - x
   \]
   
2. **Choose initial guesses**:  
   We need two points \( x_0 \) and \( x_1 \) such that \( f(x_0) \) and \( f(x_1) \) have opposite signs (i.e., the root lies between these values).  
   
   For \( f(x) = \cos(x) - x \), let's choose:
   \[
   x_0 = 0 \quad \text{and} \quad x_1 = \frac{\pi}{2} \approx 1.57
   \]
   - \( f(0) = \cos(0) - 0 = 1 \)
   - \( f(1.57) = \cos(1.57) - 1.57 \approx -1.57 \)
   
   Since \( f(0) > 0 \) and \( f(1.57) < 0 \), there is a root between \( x_0 = 0 \) and \( x_1 = 1.57 \).

3. **Apply the false position formula**:
   The next approximation \( x_2 \) is given by the formula:
   \[
   x_2 = x_1 - \frac{f(x_1)(x_1 - x_0)}{f(x_1) - f(x_0)}
   \]
   
   Substituting the values:
   \[
   x_2 = 1.57 - \frac{(-1.57)(1.57 - 0)}{-1.57 - 1}
   \]
   Simplifying:
   \[
   x_2 = 1.57 - \frac{-1.57 \times 1.57}{-2.57}
   \]
   \[
   x_2 \approx 1.57 - 0.958
   \]
   \[
   x_2 \approx 0.612
   \]

4. **Check the sign of \( f(x_2) \)**:
   \[
   f(0.612) = \cos(0.612) - 0.612 \approx 0.819 - 0.612 = 0.207
   \]
   Since \( f(x_2) > 0 \), we update \( x_0 \) to \( x_2 \) and continue the process with the interval \([0.612, 1.57]\).

5. **Iterate the process**:
   Continue applying the false position formula and refining the interval until the desired level of accuracy is achieved.

Would you like more detailed steps for further iterations, or would you like to know how to implement this numerically?

### Related Questions:
1. How does the false position method compare to the bisection method in terms of efficiency?
2. How can we ensure convergence using the false position method?
3. What are the conditions under which the false position method might fail to converge quickly?
4. What are alternative methods for finding the roots of non-linear equations, such as Newton's method?
5. How does the slope of the secant line in false position differ from Newton's method?

**Tip**: The false position method works best when the function is monotonic (strictly increasing or decreasing) between the chosen points. If the function is highly non-linear, convergence might be slow.

Learn how to approximate the root of the function f(x) = cos(x) - x using the false position method. Follow a step-by-step approach involving trigonometric functions and numerical methods. Understand key concepts like the false position formula and the intermediate value theorem, suitable for students studying Calculus or Numerical Methods.

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