Maclaurin Series for cos(x)

Maclaurin Series for $\cos(x)$

The Maclaurin series is a Taylor series centred at $x = 0$. For cosine:

$\cos(x) = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k)!} x^{2k} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$

This series converges for all real $x$.

Building Terms Iteratively

Computing each term from scratch (with a factorial and power) is slow. Instead, use the recurrence:

$t_{k+1} = t_k \cdot \frac{-x^2}{(2k+1)(2k+2)}$

Starting with $t_0 = 1$:

double maclaurin_cos(double x, int terms) {
    double sum = 0.0, term = 1.0;
    for (int k = 0; k < terms; k++) {
        sum += term;
        term *= -x * x / ((2.0*k + 1) * (2.0*k + 2));
    }
    return sum;
}

Accuracy vs. Terms

Key Values

cos(0)     = 1
cos(π/2)   = 0
cos(π)     = -1
cos(2π)    = 1

Your Task

Implement double maclaurin_cos(double x, int terms) using the iterative term recurrence.