Найти значение выражения sin(a)*sin(a)еслиa=-1/3 (синус от (a) умножить на синус от (a)еслиa равно минус 1 делить на 3) [Есть ответ!]

sin(a)*sin(a)еслиa=-1/3 (упростите выражение)

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Решение

Вы ввели [src]
sin(a)*sin(a)
$$\sin{\left(a \right)} \sin{\left(a \right)}$$
Подстановка условия [src]
sin(a)*sin(a) при a = -1/3
подставляем
sin(a)*sin(a)
$$\sin{\left(a \right)} \sin{\left(a \right)}$$
   2   
sin (a)
$$\sin^{2}{\left(a \right)}$$
переменные
a = -1/3
$$a = - \frac{1}{3}$$
   2        
sin ((-1/3))
$$\sin^{2}{\left((-1/3) \right)}$$
   2      
sin (-1/3)
$$\sin^{2}{\left(- \frac{1}{3} \right)}$$
   2     
sin (1/3)
$$\sin^{2}{\left(\frac{1}{3} \right)}$$
Степени [src]
                 2 
 /   -I*a    I*a\  
-\- e     + e   /  
-------------------
         4         
$$- \frac{\left(e^{i a} - e^{- i a}\right)^{2}}{4}$$
   2   
sin (a)
$$\sin^{2}{\left(a \right)}$$
Численный ответ [src]
sin(a)^2
Рациональный знаменатель [src]
   2   
sin (a)
$$\sin^{2}{\left(a \right)}$$
Объединение рациональных выражений [src]
   2   
sin (a)
$$\sin^{2}{\left(a \right)}$$
Общее упрощение [src]
   2   
sin (a)
$$\sin^{2}{\left(a \right)}$$
Собрать выражение [src]
   2   
sin (a)
$$\sin^{2}{\left(a \right)}$$
Общий знаменатель [src]
   2   
sin (a)
$$\sin^{2}{\left(a \right)}$$
Тригонометрическая часть [src]
/   0     for And(im(a) = 0, a mod pi = 0)
|                                         
|   1                                     
<-------             otherwise            
|   2                                     
|csc (a)                                  
\                                         
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(a \right)}} & \text{otherwise} \end{cases}$$
     1      
------------
   2/    pi\
sec |a - --|
    \    2 /
$$\frac{1}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}$$
/                        0                          for And(im(a) = 0, a mod pi = 0)
|                                                                                   
|/      0         for And(im(a) = 0, a mod pi = 0)                                  
||                                                                                  
||       2/a\                                                                       
||  4*cot |-|                                                                       
<|        \2/                                                                       
|<--------------             otherwise                         otherwise            
||             2                                                                    
||/       2/a\\                                                                     
|||1 + cot |-||                                                                     
||\        \2//                                                                     
\\                                                                                  
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}$$
   2   
sin (a)
$$\sin^{2}{\left(a \right)}$$
             2     
1     1 - tan (a)  
- - ---------------
2     /       2   \
    2*\1 + tan (a)/
$$- \frac{1 - \tan^{2}{\left(a \right)}}{2 \left(\tan^{2}{\left(a \right)} + 1\right)} + \frac{1}{2}$$
1       1     
- - ----------
2   2*sec(2*a)
$$\frac{1}{2} - \frac{1}{2 \sec{\left(2 a \right)}}$$
1   cos(2*a)
- - --------
2      2    
$$\frac{1}{2} - \frac{\cos{\left(2 a \right)}}{2}$$
/     0        for And(im(a) = 0, a mod pi = 0)
|                                              
|     1                                        
<------------             otherwise            
|   2/    pi\                                  
|sec |a - --|                                  
\    \    2 /                                  
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}$$
   1   
-------
   2   
csc (a)
$$\frac{1}{\csc^{2}{\left(a \right)}}$$
/   0     for And(im(a) = 0, a mod pi = 0)
|                                         
<   2                                     
|sin (a)             otherwise            
\                                         
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}$$
       2/a\   
  4*tan |-|   
        \2/   
--------------
             2
/       2/a\\ 
|1 + tan |-|| 
\        \2// 
$$\frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
       /pi      \
    sin|-- + 2*a|
1      \2       /
- - -------------
2         2      
$$\frac{1}{2} - \frac{\sin{\left(2 a + \frac{\pi}{2} \right)}}{2}$$
    /     1        for And(im(a) = 0, a mod pi = 0)
    |                                              
    |        2                                     
    <-1 + cot (a)                                  
    |------------             otherwise            
    |       2                                      
1   \1 + cot (a)                                   
- - -----------------------------------------------
2                          2                       
$$\frac{1}{2} - \left(\frac{\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}}{2}\right)$$
    /   1      for And(im(a) = 0, a mod pi = 0)
    <                                          
1   \cos(2*a)             otherwise            
- - -------------------------------------------
2                        2                     
$$\frac{1}{2} - \left(\frac{\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}}{2}\right)$$
/     0        for And(im(a) = 0, a mod pi = 0)
|                                              
<   2/    pi\                                  
|cos |a - --|             otherwise            
\    \    2 /                                  
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos^{2}{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}$$
/      0         for And(im(a) = 0, a mod pi = 0)
|                                                
|       2/a\                                     
|  4*cot |-|                                     
|        \2/                                     
<--------------             otherwise            
|             2                                  
|/       2/a\\                                   
||1 + cot |-||                                   
|\        \2//                                   
\                                                
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}$$
1          1       
- - ---------------
2        /pi      \
    2*csc|-- - 2*a|
         \2       /
$$\frac{1}{2} - \frac{1}{2 \csc{\left(- 2 a + \frac{\pi}{2} \right)}}$$
/      0         for And(im(a) = 0, a mod pi = 0)
|                                                
|       2/a\                                     
|  4*tan |-|                                     
|        \2/                                     
<--------------             otherwise            
|             2                                  
|/       2/a\\                                   
||1 + tan |-||                                   
|\        \2//                                   
\                                                
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}$$
   2/    pi\
cos |a - --|
    \    2 /
$$\cos^{2}{\left(a - \frac{\pi}{2} \right)}$$
/                    0                       for And(im(a) = 0, a mod pi = 0)
|                                                                            
|/   0     for And(im(a) = 0, a mod pi = 0)                                  
<|                                                                           
|<   2                                                  otherwise            
||sin (a)             otherwise                                              
\\                                                                           
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}$$
Комбинаторика [src]
   2   
sin (a)
$$\sin^{2}{\left(a \right)}$$
Раскрыть выражение [src]
   2   
sin (a)
$$\sin^{2}{\left(a \right)}$$