Aller au contenu

Mathc initiation/Fichiers c : c77cab

Un livre de Wikilivres.


Sommaire


Installer et compiler ces fichiers dans votre répertoire de travail.

c1b
/* --------------------------------- */
/* save as c1b.c                     */
/* --------------------------------- */
#include "x_hfile.h"
#include      "fb.h"
/* --------------------------------- */
int main(void)
{
double x  = 1.5;
double y  = 1.3;

 clrscrn();
 
 printf("  (x,y) = (%0.1f,%0.1f)   \n\n\n",x,y);
 
 
 printf("  %s \t\t\t= %0.8f\n", f1eq, f1(x,y));
 printf("  %s \t= %0.8f\n\n\n", f2eq, f2(x,y));
 
 stop();

 return 0;
}
/* ---------------------------------- */
/* ---------------------------------- */


Vérifions par le calcul :
  (x,y) = (1.5,1.3)   


  cos(x) + cos(y)  			            = 0.33823603
  2.* cos((x+y)/2.) *  cos((x-y)/2.) 	= 0.33823603


 Press return to continue.


Vérifions les égalités :
   posons :
     
     cos(x) + cos(y) = 2 cos( (x+y)/2 )   cos( (x-y)/2 )    
     
  Soit :  
                     = 2 cos( x/2 + y/2 ) cos( x/2 - y/2 )      
 
     Nous avons vu que :
    
     cos(x+y) = cos(x) cos(y) - sin(x) sin(y)
     cos(x-y) = cos(x) cos(y) + sin(x) sin(y)   
     
   donc  

     cos(x) + cos(y) = 2 [cos( x/2+y/2 )]
                         [cos( x/2-y/2 )]
     
                     = 2 [cos(x/2) cos(y/2) - sin(x/2) sin(y/2)]
                         [cos(x/2) cos(y/2) + sin(x/2) sin(y/2)]
                         
                     = 2 [  cos(x/2)**2 cos(y/2)**2   
                          + cos(x/2) cos(y/2) sin(x/2) sin(y/2)  
                          
                          - sin(x/2) sin(y/2) cos(x/2) cos(y/2)  
                          - sin(x/2)**2 sin(y/2)**2]                        
                         
                         
                     = 2 [  cos(x/2)**2 cos(y/2)**2  
                      
                          + cos(x/2) cos(y/2) sin(x/2) sin(y/2)                            
                          - sin(x/2) sin(y/2) cos(x/2) cos(y/2)  
                          
                          - sin(x/2)**2 sin(y/2)**2]                            
                         
                         
                     = 2 [  cos(x/2)**2 cos(y/2)**2                     
                          - sin(x/2)**2 sin(y/2)**2]                                               
                                                                        sin(x/2) = sqrt((1-cos(x))/2)
                                                                        cos(x/2) = sqrt((1+cos(x))/2)
                                                                        
                                                                        
                     = 2 [  sqrt((1+cos(x))/2)**2 sqrt((1+cos(y))/2)**2                     
                          - sqrt((1-cos(x))/2)**2 sqrt((1-cos(y))/2)**2]
                          

                     = 2 [  (1+cos(x))/2 (1+cos(y))/2                     
                          - (1-cos(x))/2 (1-cos(y))/2]                          
                          
                     = 2 [  (1+cos(x))  (1+cos(y)) /4                     
                          - (1-cos(x))  (1-cos(y)) /4]                            
                          
                     = 1/2 [  (1+cos(x))  (1+cos(y))                      
                            - (1-cos(x))  (1-cos(y)) ]     
                            
                           
                     = 1/2 [   1+cos(y)+cos(x)+cos(x)cos(y)                      
                            - (1-cos(y)-cos(x)+cos(x)cos(y)) ]                              
                                                  
     
                     = 1/2 [   1+cos(y)+cos(x)+cos(x)cos(y)                      
                            -  1+cos(y)+cos(x)-cos(x)cos(y) ]       
     
                      = 1/2 [  cos(y)+cos(x)                      
                              +cos(y)+cos(x) ]       
    
                      = 1/2 [  2cos(y)+2cos(x) ]       
     
                      =         cos(y) + cos(x)