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    <p>Factorial tends to become huge very fast (12! does not fit into a 32 bit variable); if you don't care too much about precision details and more, you can implement it <em>very</em> naively like</p> <pre><code> double fact(unsigned int n) { double r = 1.0; if (n == 0 || n == 1) return 1.0; // memoizing here can help while(n-- &gt; 0) r *= (double)(n+1); return r; } double part(double x, int n) { return (n%2 == 0? -1 : 1)*pow(x, 2*n - 1)/fact(2*n-1); } double sinx(double x) { static const double a = 0.0020202; // a number int i; double r = 0.0; for (i = 1; i &lt; MAXP; i++) r += part(x, i); return r + a; } </code></pre> <p>or something like this (I have no tested it). Of course, it is not the exact way sin can be computed correctly.</p> <hr> <p>Apart from the use of the function <em>part</em>, which holds a single "term" of the sum, this is a direct (naive) obvious translation of the mathematical formula. Let us see how to put the formula in a way that it is better to be implemented and has better performance though, to taste any difference you need to compute likely 1M sin-s or more. (This SO does not understand TeX markup as other SO sites?)</p> <pre><code>x0 = x = x/1! x1 = -x^3/3! = -x/1! * x^2/(2 * 3) = x0 * x^2/(2 * 3) x2 = x^5/5! = x^3/3! * x^2/(4 * 5) = -x1 * x^2/(4 * 5) x3 = -x^7/7! = -x^5/5! * x^2/(6 * 7) = -x2 * x^2/(6 * 7) . . . xn = -x(n-1) * x^2/(2n * (2n + 1)) for n &gt; 0 </code></pre> <p>The code given in the other answer is the translation of this "formula", not of the original one. So the code, translating this formula, would be as the other answer, more or less:</p> <pre><code>double sinx(double x, size_t t) { double term = x; double result = term; size_t i; for(i = 1; i &lt; t; i++) { term *= -x*x/( (double)(2*i*(2*i + 1)) ); result += term; } return current + trunc_err(x, t); } </code></pre> <p>where trunc_err is a function that computes the error because of the truncation of the serie. (As before, checking if everything is right is left as an exercise)</p>
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    1. POhow to Calculate trigonometric ratio using series sinx = x - (x3/3)! + (x5/5)i - (x7/7)! + a in c program
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    1. COOne objection - the utility of memoization is usually a sign of bad design. In this case it comes from badly factoring your function. If your `sinx` function needs the factorials `(2n+1)!` in sequence, it should simply keep a running variable itself, rather than forcing another function to be inefficient in either time or space to make up for not being aware of the caller's usage pattern.
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    2. COAnd likewise with your inefficient computation of powers.
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    3. CO@R.. and more: "if you don't care too much about precision details and more". I think your running variable as a special case of memoization. Why haven't you pointed out that computing sin(x) this way is anyway not the right way, and sin(x) algorithms do not use this approach at all? For the purpose of the user, this is straightforward and ok, though naif (as said)
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