Final answer:
The question involves applying Newton's method to find the root of the equation x = 2.8 cos(x) to a precision of 0.05. The method requires an initial guess and iterative calculation using the function f(x) = x - 2.8 cos(x) and its derivative.
Step-by-step explanation:
The student is asked to find the positive value of x that satisfies the equation x = 2.8 cos(x), using Newton's method. Newton's method is an iterative numerical method used for finding successively better approximations to the roots (or zeroes) of a real-valued function. The general form of Newton's method is xn+1 = xn - f(xn)/f'(xn), where f(x) is the function for which we are trying to find the root, and f'(x) is its derivative.
To apply Newton's method here, we first define f(x) as f(x) = x - 2.8 cos(x). The derivative of f(x) with respect to x is f'(x) = 1 + 2.8 sin(x). Starting with an initial guess for x (which can be taken as 0 if no better approximation is known), we apply the iterative formula to find a better approximation until the difference between successive approximations is less than 0.05, which is the desired accuracy.
It's important to remember to compute the trigonometric functions in radian mode as per the stated requirements of the problem.