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Общий знаменатель sin(pi/2+acos(3)/7)

Выражение, которое надо упростить:

Решение

Вы ввели [src]
   /pi   acos(3)\
sin|-- + -------|
   \2       7   /
$$\sin{\left(\frac{\pi}{2} + \frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}$$
sin(pi/2 + acos(3)/7)
Общее упрощение [src]
   /acos(3)\
cos|-------|
   \   7   /
$$\cos{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}$$
cos(acos(3)/7)
Разложение дроби [src]
cos(acos(3)/7)
$$\cos{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}$$
   /acos(3)\
cos|-------|
   \   7   /
Численный ответ [src]
1.03187482359459 - 2.42570912473461e-23*i
1.03187482359459 - 2.42570912473461e-23*i
Комбинаторика [src]
   /acos(3)\
cos|-------|
   \   7   /
$$\cos{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}$$
cos(acos(3)/7)
Рациональный знаменатель [src]
   /acos(3)\
cos|-------|
   \   7   /
$$\cos{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}$$
cos(acos(3)/7)
Объединение рациональных выражений [src]
   /2*acos(3) + 7*pi\
sin|----------------|
   \       14       /
$$\sin{\left(\frac{7 \pi + 2 \operatorname{acos}{\left(3 \right)}}{14} \right)}$$
sin((2*acos(3) + 7*pi)/14)
Степени [src]
   /acos(3)\
cos|-------|
   \   7   /
$$\cos{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}$$
   /     /  pi   acos(3)\      /pi   acos(3)\\ 
   |   I*|- -- - -------|    I*|-- + -------|| 
   |     \  2       7   /      \2       7   /| 
-I*\- e                   + e                / 
-----------------------------------------------
                       2                       
$$- \frac{i \left(e^{i \left(\frac{\pi}{2} + \frac{\operatorname{acos}{\left(3 \right)}}{7}\right)} - e^{i \left(- \frac{\pi}{2} - \frac{\operatorname{acos}{\left(3 \right)}}{7}\right)}\right)}{2}$$
-i*(-exp(i*(-pi/2 - acos(3)/7)) + exp(i*(pi/2 + acos(3)/7)))/2
Общий знаменатель [src]
   /acos(3)\
cos|-------|
   \   7   /
$$\cos{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}$$
cos(acos(3)/7)
Собрать выражение [src]
   /acos(3)\
cos|-------|
   \   7   /
$$\cos{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}$$
cos(acos(3)/7)
Раскрыть выражение [src]
   /acos(3)\
cos|-------|
   \   7   /
$$\cos{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}$$
cos(acos(3)/7)
Тригонометрическая часть [src]
   /acos(3)\
cos|-------|
   \   7   /
$$\cos{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}$$
     1      
------------
   /acos(3)\
sec|-------|
   \   7   /
$$\frac{1}{\sec{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}}$$
        1        
-----------------
   /pi   acos(3)\
csc|-- - -------|
   \2       7   /
$$\frac{1}{\csc{\left(\frac{\pi}{2} - \frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}}$$
        2/acos(3)\
-1 + cot |-------|
         \   14  /
------------------
       2/acos(3)\ 
1 + cot |-------| 
        \   14  / 
$$\frac{\cot^{2}{\left(\frac{\operatorname{acos}{\left(3 \right)}}{14} \right)} - 1}{\cot^{2}{\left(\frac{\operatorname{acos}{\left(3 \right)}}{14} \right)} + 1}$$
       2/acos(3)\
1 - tan |-------|
        \   14  /
-----------------
       2/acos(3)\
1 + tan |-------|
        \   14  /
$$\frac{- \tan^{2}{\left(\frac{\operatorname{acos}{\left(3 \right)}}{14} \right)} + 1}{\tan^{2}{\left(\frac{\operatorname{acos}{\left(3 \right)}}{14} \right)} + 1}$$
        /acos(3)\
 1 + sin|-------|
        \   7   /
-----------------
   /pi   acos(3)\
tan|-- + -------|
   \4       14  /
$$\frac{1 + \sin{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}}{\tan{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}$$
/       /acos(3)\\    /pi   acos(3)\
|1 + sin|-------||*cot|-- + -------|
\       \   7   //    \4       14  /
$$\left(1 + \sin{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}\right) \cot{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}$$
      /pi   acos(3)\  
 2*tan|-- + -------|  
      \4       14  /  
----------------------
       2/pi   acos(3)\
1 + tan |-- + -------|
        \4       14  /
$$\frac{2 \tan{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{1 + \tan^{2}{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}$$
      /pi   acos(3)\  
 2*cot|-- + -------|  
      \4       14  /  
----------------------
       2/pi   acos(3)\
1 + cot |-- + -------|
        \4       14  /
$$\frac{2 \cot{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{1 + \cot^{2}{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}$$
                    2                     
------------------------------------------
/            1         \    /pi   acos(3)\
|1 + ------------------|*tan|-- + -------|
|       2/pi   acos(3)\|    \4       14  /
|    tan |-- + -------||                  
\        \4       14  //                  
$$\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}\right) \tan{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}$$
                    2                     
------------------------------------------
/            1         \    /pi   acos(3)\
|1 + ------------------|*cot|-- + -------|
|       2/pi   acos(3)\|    \4       14  /
|    cot |-- + -------||                  
\        \4       14  //                  
$$\frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}\right) \cot{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}$$
/       /acos(3)\\ /       /acos(3)\\
|1 - sin|-------||*|1 + tan|-------||
\       \   7   // \       \   14  //
-------------------------------------
                  /acos(3)\          
           1 - tan|-------|          
                  \   14  /          
$$\frac{\left(1 - \sin{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}\right) \left(1 + \tan{\left(\frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}\right)}{1 - \tan{\left(\frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}$$
/       /acos(3)\\    /acos(3)\
|1 + sin|-------||*cos|-------|
\       \   7   //    \   7   /
-------------------------------
           2/pi   acos(3)\     
      2*sin |-- + -------|     
            \4       14  /     
$$\frac{\left(1 + \sin{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}\right) \cos{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}}{2 \sin^{2}{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}$$
/       /acos(3)\\ /       /acos(3)\\
|1 - tan|-------||*|1 + sin|-------||
\       \   14  // \       \   7   //
-------------------------------------
                  /acos(3)\          
           1 + tan|-------|          
                  \   14  /          
$$\frac{\left(1 + \sin{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}\right) \left(1 - \tan{\left(\frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}\right)}{1 + \tan{\left(\frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}$$
           /acos(3)\ 
      2*tan|-------| 
           \   14  / 
1 + -----------------
           2/acos(3)\
    1 + tan |-------|
            \   14  /
---------------------
     /pi   acos(3)\  
  tan|-- + -------|  
     \4       14  /  
$$\frac{1 + \frac{2 \tan{\left(\frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\tan^{2}{\left(\frac{\operatorname{acos}{\left(3 \right)}}{14} \right)} + 1}}{\tan{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}$$
/       /acos(3)\\    /  acos(3)   pi\    /pi   acos(3)\
|1 - sin|-------||*csc|- ------- + --|*sin|-- + -------|
\       \   7   //    \     14     4 /    \4       14  /
$$\left(1 - \sin{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}\right) \sin{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)} \csc{\left(\frac{\pi}{4} - \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}$$
/       /acos(3)\\    /pi   acos(3)\    /  acos(3)   pi\
|1 + sin|-------||*csc|-- + -------|*sin|- ------- + --|
\       \   7   //    \4       14  /    \     14     4 /
$$\left(1 + \sin{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}\right) \sin{\left(\frac{\pi}{4} - \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)} \csc{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}$$
/           /acos(3)\ \                  
|      2*cot|-------| |                  
|           \   14  / |    /pi   acos(3)\
|1 + -----------------|*cot|-- + -------|
|           2/acos(3)\|    \4       14  /
|    1 + cot |-------||                  
\            \   14  //                  
$$\left(1 + \frac{2 \cot{\left(\frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\cot^{2}{\left(\frac{\operatorname{acos}{\left(3 \right)}}{14} \right)} + 1}\right) \cot{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}$$
/       /acos(3)\\    /pi   acos(3)\
|1 + sin|-------||*cos|-- + -------|
\       \   7   //    \4       14  /
------------------------------------
           /  pi   acos(3)\         
        cos|- -- + -------|         
           \  4       14  /         
$$\frac{\left(1 + \sin{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}\right) \cos{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\cos{\left(- \frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}$$
/       /acos(3)\\    /  pi   acos(3)\
|1 - sin|-------||*cos|- -- + -------|
\       \   7   //    \  4       14  /
--------------------------------------
             /pi   acos(3)\           
          cos|-- + -------|           
             \4       14  /           
$$\frac{\left(1 - \sin{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}\right) \cos{\left(- \frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\cos{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}$$
/         1      \    /pi   acos(3)\
|1 + ------------|*csc|-- + -------|
|       /acos(3)\|    \4       14  /
|    csc|-------||                  
\       \   7   //                  
------------------------------------
           /  acos(3)   pi\         
        csc|- ------- + --|         
           \     14     4 /         
$$\frac{\left(1 + \frac{1}{\csc{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}}\right) \csc{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\csc{\left(\frac{\pi}{4} - \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}$$
/       /  pi   acos(3)\\    /pi   acos(3)\
|1 + cos|- -- + -------||*cos|-- + -------|
\       \  2       7   //    \4       14  /
-------------------------------------------
               /  pi   acos(3)\            
            cos|- -- + -------|            
               \  4       14  /            
$$\frac{\left(1 + \cos{\left(- \frac{\pi}{2} + \frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}\right) \cos{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\cos{\left(- \frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}$$
/             1         \    /  pi   acos(3)\
|1 + -------------------|*sec|- -- + -------|
|       /  pi   acos(3)\|    \  4       14  /
|    sec|- -- + -------||                    
\       \  2       7   //                    
---------------------------------------------
                 /pi   acos(3)\              
              sec|-- + -------|              
                 \4       14  /              
$$\frac{\left(1 + \frac{1}{\sec{\left(- \frac{\pi}{2} + \frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}}\right) \sec{\left(- \frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\sec{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}$$
                    /acos(3)\                
                 cos|-------|                
                    \   7   /                
---------------------------------------------
/          2/acos(3)\    \                   
|       cos |-------|    |                   
|           \   7   /    |    2/pi   acos(3)\
|1 + --------------------|*sin |-- + -------|
|         4/pi   acos(3)\|     \4       14  /
|    4*sin |-- + -------||                   
\          \4       14  //                   
$$\frac{\cos{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}}{4 \sin^{4}{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}\right) \sin^{2}{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}$$
               2/pi   acos(3)\         
          4*sin |-- + -------|         
                \4       14  /         
---------------------------------------
/         4/pi   acos(3)\\             
|    4*sin |-- + -------||             
|          \4       14  /|    /acos(3)\
|1 + --------------------|*cos|-------|
|          2/acos(3)\    |    \   7   /
|       cos |-------|    |             
\           \   7   /    /             
$$\frac{4 \sin^{2}{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\left(1 + \frac{4 \sin^{4}{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\cos^{2}{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}}\right) \cos{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}}$$
            /       /acos(3)\\         
          2*|1 + sin|-------||         
            \       \   7   //         
---------------------------------------
/         4/pi   acos(3)\\             
|    4*sin |-- + -------||             
|          \4       14  /|    /acos(3)\
|1 + --------------------|*cos|-------|
|          2/acos(3)\    |    \   7   /
|       cos |-------|    |             
\           \   7   /    /             
$$\frac{2 \cdot \left(1 + \sin{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}\right)}{\left(1 + \frac{4 \sin^{4}{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\cos^{2}{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}}\right) \cos{\left(\frac{\operatorname{acos}{\left(3 \right)}}{7} \right)}}$$
                  /pi   acos(3)\              
             2*csc|-- + -------|              
                  \4       14  /              
----------------------------------------------
/        2/pi   acos(3)\ \                    
|     csc |-- + -------| |                    
|         \4       14  / |    /  acos(3)   pi\
|1 + --------------------|*csc|- ------- + --|
|       2/  acos(3)   pi\|    \     14     4 /
|    csc |- ------- + --||                    
\        \     14     4 //                    
$$\frac{2 \csc{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\left(1 + \frac{\csc^{2}{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\csc^{2}{\left(\frac{\pi}{4} - \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}\right) \csc{\left(\frac{\pi}{4} - \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}$$
                /  acos(3)   pi\            
           2*csc|- ------- + --|            
                \     14     4 /            
--------------------------------------------
/       2/  acos(3)   pi\\                  
|    csc |- ------- + --||                  
|        \     14     4 /|    /pi   acos(3)\
|1 + --------------------|*csc|-- + -------|
|        2/pi   acos(3)\ |    \4       14  /
|     csc |-- + -------| |                  
\         \4       14  / /                  
$$\frac{2 \csc{\left(\frac{\pi}{4} - \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\left(1 + \frac{\csc^{2}{\left(\frac{\pi}{4} - \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\csc^{2}{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}\right) \csc{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}$$
                /  pi   acos(3)\            
           2*cos|- -- + -------|            
                \  4       14  /            
--------------------------------------------
/       2/  pi   acos(3)\\                  
|    cos |- -- + -------||                  
|        \  4       14  /|    /pi   acos(3)\
|1 + --------------------|*cos|-- + -------|
|        2/pi   acos(3)\ |    \4       14  /
|     cos |-- + -------| |                  
\         \4       14  / /                  
$$\frac{2 \cos{\left(- \frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\left(1 + \frac{\cos^{2}{\left(- \frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\cos^{2}{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}\right) \cos{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}$$
                  /pi   acos(3)\              
             2*cos|-- + -------|              
                  \4       14  /              
----------------------------------------------
/        2/pi   acos(3)\ \                    
|     cos |-- + -------| |                    
|         \4       14  / |    /  pi   acos(3)\
|1 + --------------------|*cos|- -- + -------|
|       2/  pi   acos(3)\|    \  4       14  /
|    cos |- -- + -------||                    
\        \  4       14  //                    
$$\frac{2 \cos{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\cos^{2}{\left(- \frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}\right) \cos{\left(- \frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}$$
                /  pi   acos(3)\            
           2*sec|- -- + -------|            
                \  4       14  /            
--------------------------------------------
/       2/  pi   acos(3)\\                  
|    sec |- -- + -------||                  
|        \  4       14  /|    /pi   acos(3)\
|1 + --------------------|*sec|-- + -------|
|        2/pi   acos(3)\ |    \4       14  /
|     sec |-- + -------| |                  
\         \4       14  / /                  
$$\frac{2 \sec{\left(- \frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\left(1 + \frac{\sec^{2}{\left(- \frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\sec^{2}{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}\right) \sec{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}$$
                  /pi   acos(3)\              
             2*sec|-- + -------|              
                  \4       14  /              
----------------------------------------------
/        2/pi   acos(3)\ \                    
|     sec |-- + -------| |                    
|         \4       14  / |    /  pi   acos(3)\
|1 + --------------------|*sec|- -- + -------|
|       2/  pi   acos(3)\|    \  4       14  /
|    sec |- -- + -------||                    
\        \  4       14  //                    
$$\frac{2 \sec{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}{\sec^{2}{\left(- \frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}\right) \sec{\left(- \frac{\pi}{4} + \frac{\operatorname{acos}{\left(3 \right)}}{14} \right)}}$$
2*sec(pi/4 + acos(3)/14)/((1 + sec(pi/4 + acos(3)/14)^2/sec(-pi/4 + acos(3)/14)^2)*sec(-pi/4 + acos(3)/14))