Question

\(x^{2}-2 x+1=\sin x\)

Short answer

The exact solution of the problem can’t be expressed with elementary functions. The approximate solution can be found using the Newton-Raphson method or other similar numerical methods.

Step-by-step solution

Understand the Newton-Raphson method

The Newton-Raphson method is used to find successively better approximations to the roots (or zeroes) of a real-valued function. The process starts with a function \(f\) defined over the real numbers \(x\), and its derivative \(f'\), an initial guess \(x_0\) for a root of the function \(f\). Then a new possible root \(x_1\) is calculated using the formula: \(x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}\). The process repeats until a sufficiently accurate value is reached.

Define the function and its derivative

The given equation is \(f(x)= x^2 -2x +1 - \sin(x) = 0\). Therefore, the derivative of this function \(f'(x)\) is \(2x -2 - \cos(x)\).

Make an initial guess

There is no hard and fast rule for guessing the initial value of \(x_0\). However, by plotting the function, we can make an intelligent guess. Here, let's take \(x_0 = 0.5\).

Apply the Newton-Raphson method

Use the Newton-Raphson formula to calculate \(x_1\). Then, use \(x_1\) to calculate \(x_2\), and so on, until the desired level of precision is reached. This is done by looping over the formula: \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\).