This C program computes the value of cos(x) by summing its Taylor series up to a given accuracy, then checks the result against the built-in cos() library function. It’s a great exercise in loops, floating-point arithmetic, and how mathematical functions are actually approximated inside a computer.

The Cosine Series

The Taylor series for cosine (with x in radians) is:

cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + ...

Each term is built from the previous one by multiplying by -x² / (2n(2n-1)), so we never have to compute factorials or powers directly — we just keep refining the sum until it is close enough to the library value.

The Program

#include <stdio.h>
#include <math.h>

#define PI 3.14159265358979323846

int main(void)
{
    int n, deg;
    double acc, term, x, cosx, cosval;

    printf("Enter the value of x (in degrees) : ");
    scanf("%d", &deg);

    printf("Enter the accuracy for the result : ");
    scanf("%lf", &acc);

    x = deg * (PI / 180.0);   /* convert degrees to radians */
    cosval = cos(x);          /* reference value from the library */

    term = 1.0;               /* first term of the series */
    cosx = term;
    n = 1;

    do {
        term = -term * x * x / (2.0 * n * (2.0 * n - 1.0));
        cosx += term;
        n++;
    } while (fabs(cosval - cosx) >= acc);

    printf("Sum of the cosine series       = %lf\n", cosx);
    printf("Using library function cos(%d) = %lf\n", deg, cosval);
    return 0;
}

How the Program Works

• The angle is entered in degrees and converted to radians (x = deg * PI / 180), because C’s cos() expects radians.
• term starts at 1 (the first series term). Each iteration multiplies it by -x² / (2n(2n-1)) to get the next term and adds it to cosx.
• The loop keeps adding terms until the running sum is within acc of the library’s cos(x) — that is how we control accuracy.
• We use double (not float) and a full-precision value of π for an accurate result, improving on the old 3.142 constant.

Compiling (link the math library)

On Linux and macOS you must link the math library with -lm:

gcc cosx.c -o cosx -lm

Sample Output

Enter the value of x (in degrees) : 60
Enter the accuracy for the result : 0.00001
Sum of the cosine series       = 0.500000
Using library function cos(60) = 0.500000

For a solid grounding in C’s numeric types and the math library, The C Programming Language by Kernighan and Ritchie remains the classic reference — find it on Amazon.

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