Final answer:
Newton's method involves iteratively updating the root approximation of an equation using the derivative until the difference between successive approximations is within a desired tolerance. For the equation x = 0.8 cos(x), you transform it into f(x) = x - 0.8 cos(x), and apply the method starting with an initial guess, continuing until the changes are minimal.
Step-by-step explanation:
To solve the equation x = 0.8 cos(x) using Newton's method, we first need to reframe it into a form suitable for the application of the method. We are looking for a root of the equation f(x) = x - 0.8 cos(x), which means we want to find an x such that f(x) = 0. Newton's method updates the guess for the root according to the formula xn+1 = xn - f(xn)/f'(xn), where f'(x) is the derivative of f(x).
The derivative of our function is f'(x) = 1 + 0.8 sin(x). We can choose a reasonable starting point, like x0 = 1 because the cosine of values close to 1 is also close to 1, which makes x = 0.8 cos(x) conceivable. Using Newton's method, we recursively apply the formula for new approximations until the change |xn+1 - xn| is within the desired tolerance of 0.05.
By performing the iterations, you will eventually reach a stage where the absolute difference between successive approximations is less than 0.05, at which point you can stop and accept the current approximation as the positive value of x that satisfies the equation within the required accuracy.