In the last post I discussed and showed you how to write a program that finds the sum of the Sine series(Infinite so to speak).

Using the same concept, we will extend it a bit further in this post.

In this post we will evaluate the Cosine series, correct upto a certain decimal places, for a given range of x in radians. We would store the value of Cos(x) evaluated in a text file and then plot them using Gnuplot.

So let’s first start with writing a program that evaluates the Cosine series.

The series is given by:

Ratio of the successive terms(index the numbers from 0):

Since, we indexed the terms starting from 0, therefore, for the above relation to work, will go from 1 to .

Now, knowing the first() term, the successive terms can be calculated as :

and so on.

Therefore, the C program that calculates the sum of the cosine series upto a certain accuracy can be written as shown below.

### PROGRAM:

/******************************** ******INFINITE SERIES SUM********** Series: cos(x) = 1 - (x^2/2!) + (x^4/4!) - ..... ********************************/ #include<stdio.h> #include<math.h> main(){ int i=1; double x,t0,t1,R,sum,eps; printf("Enter the value of x:\n"); scanf("%lf",&x); printf("Enter the desired accuracy: "); scanf("%lf",&eps); //Initialize First Term t0=1; //Make sum equal to the first term sum=t0; do{ //Find the ratio of the second term to the first term using already known relation R=-(x*x)/(2*i-1)/(2*i); //Calculate the second term t1=R*t0; //find the new sum sum=sum+t1; t0=t1; i++; //keep on summing terms until the required accuracy is reached }while(fabs(t1/sum)>eps); printf("\nThe sum [cos(%lf)] is: %lf with %d terms",x,sum,i); }

### OUTPUT:

The program asks the user to enter the value of x and the desired accuracy, and gives answer.

Now that we have a program for evaluating the cosine series, we can write a program that will evaluate the cosine series in a given range[0 to 4pi] and store the values in a file.

### PROGRAM:

/******************************** ******INFINITE SERIES SUM********** Series: cos(x) = 1 - (x^2/2!) + (x^4/4!) - ..... ********************************/ #include<stdio.h> #include<math.h> main(){ FILE *fp=NULL; fp=fopen("cos(x).txt","w"); double x,t0,t1,R,sum,eps; printf("Enter the desired accuracy: "); scanf("%lf",&eps); for(x=0.0000;x<=4*M_PI;x=x+0.001){ int i=1; //Initialize First Term t0=1; //Make sum equal to the first term sum=t0; do{ //Find the ratio of the second term to the first term using already known relation R=-(x*x)/(2*i-1)/(2*i); //Calculate the second term t1=R*t0; //find the new sum sum=sum+t1; t0=t1; i++; //keep on summing terms until the required accuracy is reached }while(fabs(t1/sum)>eps); fprintf(fp,"%lf\t%lf\n",x,sum); } }

### OUPTUT:

When you run the above C program it will ask for the accuracy desired for the calculations.

When the execution is complete, it will create a txt file called ‘cos(x).txt’ that will contain the data to be plotted.

### Gnuplot Command:

You can plot the data using Gnuplot, by giving the following command:

`plot './cos(x).txt' w l`

### Gnuplot OUPTUT:

Ph.D. researcher at Friedrich-Schiller University Jena, Germany. I’m a physicist specializing in computational material science. I write efficient codes for simulating light-matter interactions at atomic scales. I like to develop Physics, DFT, and Machine Learning related apps and software from time to time. Can code in most of the popular languages. I like to share my knowledge in Physics and applications using this Blog and a YouTube channel.