Question

Write an overloaded procedure using c++ for approximating the value of cos(x), called approx_cos. This procedure should work as follows:

• A call such as approx_cos(double x, long N) should produce the Nth order Taylor approximation of the cosine function at the point x. You of course need the derive this Taylor expansion in order to implement it.

• A call such as approx_cos(double x) should approximate the value of the cosine function evaluated at the point x, to at least eight digits of accuracy. The Taylor Remainder Theorem helps us understand how to mathematically obtain such a result.

Of course you should write a main() procedure that checks the validity of this procedure and compare it to the true value of cosine at several points (not just x=0).

Answer 1

#include <iostream>
#include <cmath>

using namespace std;

double approx_cos(double x) {
        double lastResult = 0;
        double result = 1;
        double lastTerm = 1;
        int i=1;
        while(abs(result - lastResult) > pow(10, -6)) {
                lastResult = result;
                lastTerm = -1 * lastTerm * (x * x)/(2*i * (2*i - 1));
                result += lastTerm;
                i++;
        }

        return result;
}

double cosine(double x, long n) {

        double result = 1;
        double lastTerm = 1;
        for(int i=1; i<=n; i++) {
                lastTerm = -1 * lastTerm * (x * x)/(2*i * (2*i - 1));
                result += lastTerm;
        }

        return result;
}

int main() {

        cout << cos(M_PI/8.0) << endl;
        cout << cosine(M_PI/8.0, 100) << endl;
        cout << approx_cos(M_PI/8.0) << endl;

}

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