Code:/* A program to compute the sine of an angle using sine's series expansion and without the use of the maths library David Tims 10319139 timsd@tcd.ie */ #include <stdio.h> double dfactorial(int n); /*function prototype for dfactorial (given n, outputs n!), mypow (given x and n outputs x^n) and mysine (given x and nmax computes sin(x) for nmax the iterating value) */ double mypow(double x, int n); double mysine(double x, int nmax); int main() { double x; int nmax; printf("Enter a value for 'x' between (pi, -pi) for which 'sin(x)' is to evaluated\n and an iterating value 'n'\n"); /*asks user for values of 'x' and 'nmax' */ scanf("%lf" "%d", &x, &nmax); mysine(x,nmax); /* applies mysine on 'x' and 'nmax' prducing an approx to sin(x) */ printf("sin(x)=%lf\n", mysine(x,nmax)); /* prints the approx found by mysine for sin(x) */ return 0; /* lets compiler know program has reached the end */ } double dfactorial(int n) /* defining ifactorial */ { double i;/*declaring i and z, two new variables to use in ifactorial */ double z; i=1.0; /* defining i and z */ z=1.0; if(n==0) /* For n=o the factorial is mathematically defined as 1 */ { return(1.0); /* returns 1 */ } else if (n>0) /* for positive n computes algorithm for factorial */ { while(i<=n) { z=z*i; /* e.g case n=2: z=1*1=1 update i++ so i=1+1=2 ===> update z, z=1*2=2!=n! etc for general n */ i++; } return(z); /* returns final value of z i.e n! */ } else { printf("You need to enter a positive integer, disregard all output and try again\n"); /* Just to let the user know n needs to be positive and the output is useless */ return(0); } } double mypow(double x, int n) /* defining "mypow" */ { int a; /* declaring variables a, b and y to use in mypow */ double b; double y; y=1/x; /* need this for the case where n is negative ( i.e x^(-n) = 1/x^n = 1/x*....*1/x ) */ a=1.0; b=1.0; if(n==0) /* if n is 0, by definition the value of x^n is 1 (mathematically defined) */ { return(1); } else if (n>0)/* computes algorithm for x^n */ { while(a<=n) { b=b*x; /* e.g n=2 case: b=1*x=x, update a: a=1+1=2 ===> b=x*x=x^2=x^n etc for gen n */ a++; } return(b); /* returns final value of z, which is the x^n value required */ } else /* computes algorithm for x^n when n is negative, note because n is now negative the condition for i needed to be adjusted */ { while(a>=n+2) { b=b*y; a--; } return(b); } } double mysine(double x, int nmax) /*defining mysine function, note it calls other functions for each iteration of n */ { double c; /* daclaring variables c and t to use in mysine */ double t; int n; c=0; for (n=0; n<nmax; n++) { t = (mypow(-1,n))*(x)*(mypow(x,n))*(mypow(x,n))*(1/(dfactorial(2*n+1))); /* t is the nth tern in the sine series expansion */ c=c+t; /*updates the sum after each value of n to include nth term */ n++; } return c; }