To apply Newton's method to find an approximate solution to the equation

$e^{1.6x} = 1.3 + 1.6 \cos^2(x),$

we'll first rewrite this equation in the form suitable for Newton's method, which requires the equation to be set to zero:

$f(x) = e^{1.6x} - 1.3 - 1.6 \cos^2(x).$

Newton's method uses the following iteration formula:

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

where $f'(x)$ is the derivative of $f(x)$.

Step 1: Calculate $f(x)$ and $f'(x)$

First, compute the derivative $f'(x)$:

$f'(x) = \frac{d}{dx} \left(e^{1.6x} - 1.3 - 1.6 \cos^2(x) \right).$

This simplifies to:

$f'(x) = 1.6e^{1.6x} + 1.6 \cdot 2 \cos(x) (-\sin(x)) = 1.6e^{1.6x} - 3.2 \cos(x) \sin(x).$

Thus,

$f'(x) = 1.6e^{1.6x} - 3.2 \cos(x) \sin(x).$

Step 2: Apply Newton's Method

We start with an initial estimate $x_0 = 1.07$.

Now, let's compute three iterations of Newton's method. I'll calculate the values step-by-step.After applying three iterations of Newton's method, the approximations for the solution are:

1. $x_1 \approx 0.55496$
2. $x_2 \approx 0.56542$
3. $x_3 \approx 0.55329$

These values are successive approximations that converge towards the solution of the equation.

Would you like a detailed explanation of each iteration, or do you have any other questions?

Further Exploration Questions:

1. How does Newton's method work in terms of convergence speed?
2. What would happen if we chose a different initial estimate $x_0$?
3. Can Newton's method fail to converge, and under what conditions?
4. How can we determine the accuracy of the approximate solution obtained?
5. What are some alternative methods for solving nonlinear equations numerically?

Tip: Always choose an initial estimate close to the expected root for faster convergence in Newton's method.