sin(x) and cos(x) Using Taylor Series in C

Algorithm

1. Read the angle x in radians.
2. Compute sin(x) = x - x³/3! + x⁵/5! - ..., building each term from the previous one.
3. Compute cos(x) = 1 - x²/2! + x⁴/4! - ..., again building each term from the previous one.
4. Print the Taylor values next to the math.h values.
5. If cos(x) is zero, report that the ratio is undefined; otherwise compute sin(x)/cos(x) both ways and compare them.

C program

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

#define TERMS 15
#define ZERO_TOLERANCE 1e-12

double taylor_sin(double x) {
    double term = x;
    double sum = x;
    int k;

    for (k = 1; k <= TERMS; k++) {
        term *= -x * x / ((2 * k) * (2 * k + 1));
        sum += term;
    }
    return sum;
}

double taylor_cos(double x) {
    double term = 1.0;
    double sum = 1.0;
    int k;

    for (k = 1; k <= TERMS; k++) {
        term *= -x * x / ((2 * k - 1) * (2 * k));
        sum += term;
    }
    return sum;
}

int main(void) {
    double x, s, c, ratio_taylor, ratio_lib, diff;

    printf("Enter angle x in radians:\n");
    if (scanf("%lf", &x) != 1) {
        printf("Error: invalid input\n");
        return 1;
    }

    s = taylor_sin(x);
    c = taylor_cos(x);
    printf("%-26s: %.6f\n", "sin(x) by Taylor series", s);
    printf("%-26s: %.6f\n", "sin(x) from math.h", sin(x));
    printf("%-26s: %.6f\n", "cos(x) by Taylor series", c);
    printf("%-26s: %.6f\n", "cos(x) from math.h", cos(x));

    if (fabs(c) < ZERO_TOLERANCE) {
        printf("sin(x)/cos(x) is undefined because cos(x) is zero\n");
        return 0;
    }

    ratio_taylor = s / c;
    ratio_lib = sin(x) / cos(x);
    diff = fabs(ratio_taylor - ratio_lib);
    printf("%-26s: %.6f\n", "sin(x)/cos(x) by Taylor", ratio_taylor);
    printf("%-26s: %.6f\n", "sin(x)/cos(x) from math.h", ratio_lib);

    if (diff < 1e-6) {
        printf("Inference: the Taylor result agrees with math.h to 6 decimal places\n");
    } else {
        printf("Inference: the results differ by %.6f\n", diff);
    }
    return 0;
}

Sample input and output

0.5

Enter angle x in radians:
sin(x) by Taylor series   : 0.479426
sin(x) from math.h        : 0.479426
cos(x) by Taylor series   : 0.877583
cos(x) from math.h        : 0.877583
sin(x)/cos(x) by Taylor   : 0.546302
sin(x)/cos(x) from math.h : 0.546302
Inference: the Taylor result agrees with math.h to 6 decimal places

0

Enter angle x in radians:
sin(x) by Taylor series   : 0.000000
sin(x) from math.h        : 0.000000
cos(x) by Taylor series   : 1.000000
cos(x) from math.h        : 1.000000
sin(x)/cos(x) by Taylor   : 0.000000
sin(x)/cos(x) from math.h : 0.000000
Inference: the Taylor result agrees with math.h to 6 decimal places

How the program works

Each Taylor term is derived from the previous one by multiplying by -x² and dividing by the next two factorial factors. This avoids computing large factorials and powers separately, which would overflow quickly.

Fifteen terms are plenty for angles of a few radians, so the Taylor values match math.h to six decimal places. For large x the series converges more slowly, which is why the lab asks for an inference rather than an exact match.

Viva questions

Common mistakes

• Using integer division inside the term formula, which turns the terms into zero.
• Forgetting that math.h functions take radians, not degrees.
• Comparing the two floating-point results with == instead of a tolerance.