We are tasked with applying Newton's method to find the second and third approximations for the equation $3\sin(x) = 3$, starting from the initial approximation $x_1 = 1$.

Step 1: Rearrange the equation

First, rewrite the given equation in a standard form suitable for Newton's method, where we need the equation in the form $f(x) = 0$.

$3\sin(x) = 3 \implies \sin(x) = 1$

Now define the function $f(x)$ as:

$f(x) = \sin(x) - 1$

Step 2: Newton's Method Formula

Newton's method iteratively improves the approximation to the root by using the formula:

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Where:

• $f(x) = \sin(x) - 1$
• $f'(x) = \cos(x)$ (the derivative of $f(x)$)

Step 3: Calculate the second approximation $x_2$

We are given that $x_1 = 1$.

1. Compute $f(x_1)$:

$f(x_1) = \sin(1) - 1 \approx 0.84147 - 1 = -0.15853$

2. Compute $f'(x_1)$:

$f'(x_1) = \cos(1) \approx 0.5403$

3. Use Newton's formula to find $x_2$:

$x_2 = x_1 - \frac{f(x_1)}{f'(x_1)} = 1 - \frac{-0.15853}{0.5403} \approx 1 + 0.29346 \approx 1.2935$

Step 4: Calculate the third approximation $x_3$

Now, repeat the process using $x_2 = 1.2935$.

1. Compute $f(x_2)$:

$f(x_2) = \sin(1.2935) - 1 \approx 0.96136 - 1 = -0.03864$

2. Compute $f'(x_2)$:

$f'(x_2) = \cos(1.2935) \approx 0.2756$

3. Use Newton's formula to find $x_3$:

$x_3 = x_2 - \frac{f(x_2)}{f'(x_2)} = 1.2935 - \frac{-0.03864}{0.2756} \approx 1.2935 + 0.1402 \approx 1.4337$

Final Result:

• Second approximation $x_2 \approx 1.2935$
• Third approximation $x_3 \approx 1.4337$

Would you like more details or have any questions?

5 Related Questions:

1. How does Newton's method converge to the root of a function?
2. What happens if the derivative $f'(x)$ is close to zero in Newton's method?
3. Can Newton's method fail? If so, under what conditions?
4. How does Newton's method compare to other root-finding algorithms like bisection or secant method?
5. What are the limitations of using Newton's method for non-linear equations?

Tip:

Newton's method is very sensitive to the initial guess. A poor choice of starting point may result in divergence or slow convergence.