Use Newton's method to solve the equation sec x = 4 in the interval x in (0, pi/2).
In other words, use Newton's Method to compute arcsec(4).
(You need to make a good initial guess for the root otherwise Newton's method will probably fail. Please justify how you came up with your initial guess. Stop the iteration process
when you are correct to 4 decimal places.)
You need to be doing this on your calculator.
I will start you.
x=75 deg
sec75=3.36
try higher
sec 78=4.8 lower
sec 76.5=4.28 lower
sec 76=4.13 lower
sec 75.5=3.99 higher
sec 75.51=3.9966 higher
sec 75.515=3.99797 higher
sec 75.52=3.9993
sec 75.525=4.00067 lower
sec 75.524=4.0004
sec 75.523=4.0001
sec 75.5228=4.00008
sec 75.5223=3.99995=
sec 75.5225=4.000003
with practice, you get pretty good.
thanks but i need to use newton's method for this.... i tried it by starting with 1 but do not get an answer. :s
To solve the equation sec(x) = 4 using Newton's method, we can follow these steps:
1. First, let's rewrite the equation in terms of cos(x): cos(x) = 1/4. This equation will be our target equation to find the root.
2. Newton's method requires an initial guess for the root. In this case, we need an initial guess for x in the interval (0, pi/2). To come up with a good initial guess, it is helpful to make a graph of the function cos(x) - 1/4 and visually estimate where the root might be in the interval (0, pi/2).
By observing the graph, we can see that the root is close to x = pi/3 (approximately 1.0472 radians). Therefore, we will choose x0 = 1.0472 as our initial guess.
3. Let's denote the current estimate of the root as xn, and the next estimate as xn+1. We will repeat the iteration process until we reach the desired accuracy.
The formula for Newton's method is: xn+1 = xn - f(xn) / f'(xn), where f(x) is the target equation and f'(x) is its derivative.
4. Differentiating cos(x) - 1/4 with respect to x, we get -sin(x). Therefore, f'(x) = -sin(x).
5. Now, we can start the iteration process using the initial guess x0 = 1.0472:
- Calculate f(x0) = cos(x0) - 1/4 and f'(x0) = -sin(x0).
- Substitute these values into the Newton's method formula to find x1: x1 = x0 - f(x0) / f'(x0).
- Repeat the process using x1 as the new value of xn until we reach the desired accuracy.
6. Iterating through these steps, we will eventually converge to the root with the desired precision.
It is important to note that Newton's method may fail if the initial guess is not sufficiently close to the root or if we encounter a singularity.
I will now perform the calculations using Newton's method to compute the value of arcsec(4) accurate to 4 decimal places.