To solve this equation using Newton's method, let's break down the process step-by-step.

The equation provided is: $3x^2 - 6x + 3 = \sin(x)$

To apply Newton's method, we need to rewrite it in the form $f(x) = 0$. Let's set: $f(x) = 3x^2 - 6x + 3 - \sin(x)$

Step 1: Compute $f'(x)$

To use Newton's method, we need the derivative $f'(x)$: $f'(x) = 6x - 6 - \cos(x)$

Step 2: Newton's Method Formula

The iterative formula for Newton's method is: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Step 3: Choose an Initial Guess

Choose a starting point $x_0$. It appears that $x = 1.983680$ might be the solution provided. To verify or find any additional solutions, we can iterate from different initial guesses within the function's reasonable domain.

Step 4: Iterative Computation

For each initial guess $x_0$, apply the formula iteratively until the result converges to a value that does not change (within six decimal places).

If you'd like, I can walk through the calculations in more detail or perform additional iterations if you want to verify the provided solution or check for other possible solutions.

Would you like the detailed iterative steps or have any questions about this process?

Here are five follow-up questions for further exploration:

1. What is the significance of choosing a good initial guess in Newton's method?
2. How does the convergence of Newton's method depend on the properties of $f(x)$ and $f'(x)$?
3. What are some potential limitations of using Newton's method for finding roots?
4. How does the presence of a trigonometric function $\sin(x)$ affect the application of Newton's method in this problem?
5. Could other numerical methods, like the Secant Method, be used for this equation, and how would they compare?

Tip: Always verify if multiple roots exist by selecting different initial guesses in Newton's method, as some functions can have multiple real roots within the same interval.