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

Вы ввели

sin(pi - 3*a)

$$\sin{\left(\pi - 3 a \right)}$$

Подстановка условия

sin(pi - 3*a) при a = -1/3

подставляем

sin(pi - 3*a)

$$\sin{\left(\pi - 3 a \right)}$$

sin(3*a)

$$\sin{\left(3 a \right)}$$

переменные

a = -1/3

$$a = - \frac{1}{3}$$

sin(3*(-1/3))

$$\sin{\left(3 (-1/3) \right)}$$

-sin(1)

$$- \sin{\left(1 \right)}$$

Степени

sin(3*a)

$$\sin{\left(3 a \right)}$$

   /   I*(-pi + 3*a)    I*(pi - 3*a)\ 
-I*\- e              + e            / 
--------------------------------------
                  2                   

$$- \frac{i \left(e^{i \left(\pi - 3 a\right)} - e^{i \left(3 a - \pi\right)}\right)}{2}$$

Численный ответ

sin(pi - 3*a)

Рациональный знаменатель

sin(3*a)

$$\sin{\left(3 a \right)}$$

Объединение рациональных выражений

sin(3*a)

$$\sin{\left(3 a \right)}$$

Общее упрощение

sin(3*a)

$$\sin{\left(3 a \right)}$$

Собрать выражение

sin(3*a)

$$\sin{\left (3 a \right )}$$

Комбинаторика

sin(3*a)

$$\sin{\left(3 a \right)}$$

Общий знаменатель

sin(3*a)

$$\sin{\left(3 a \right)}$$

Тригонометрическая часть

      1      
-------------
   /      pi\
sec|3*a - --|
   \      2 /

$$\frac{1}{\sec{\left(3 a - \frac{\pi}{2} \right)}}$$

sin(3*a)

$$\sin{\left(3 a \right)}$$

/      0        for And(im(a) = 0, -3*a mod pi = 0)
|                                                  
|       /3*a\                                      
|  2*tan|---|                                      
<       \ 2 /                                      
|-------------               otherwise             
|       2/3*a\                                     
|1 + tan |---|                                     
\        \ 2 /                                     

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge - 3 a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{3 a}{2} \right)}}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}$$

/      0        for And(im(a) = 0, 3*a mod pi = 0)
|                                                 
|       /3*a\                                     
|  2*cot|---|                                     
<       \ 2 /                                     
|-------------              otherwise             
|       2/3*a\                                    
|1 + cot |---|                                    
\        \ 2 /                                    

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}$$

/                        0                           for And(im(a) = 0, -3*a mod pi = 0)
|                                                                                       
|/      0        for And(im(a) = 0, 3*a mod pi = 0)                                     
||                                                                                      
||       /3*a\                                                                          
<|  2*cot|---|                                                                          
|<       \ 2 /                                                    otherwise             
||-------------              otherwise                                                  
||       2/3*a\                                                                         
||1 + cot |---|                                                                         
\\        \ 2 /                                                                         

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge - 3 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}$$

/      0        for And(im(a) = 0, -3*a mod pi = 0)
|                                                  
<   /      pi\                                     
|cos|3*a - --|               otherwise             
\   \      2 /                                     

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge - 3 a \bmod \pi = 0 \\\cos{\left(3 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}$$

      1      
-------------
   /pi      \
sec|-- - 3*a|
   \2       /

$$\frac{1}{\sec{\left(- 3 a + \frac{\pi}{2} \right)}}$$

   1    
--------
csc(3*a)

$$\frac{1}{\csc{\left(3 a \right)}}$$

       /3*a\ 
  2*tan|---| 
       \ 2 / 
-------------
       2/3*a\
1 + tan |---|
        \ 2 /

$$\frac{2 \tan{\left(\frac{3 a}{2} \right)}}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1}$$

/   0      for And(im(a) = 0, -3*a mod pi = 0)
<                                             
\sin(3*a)               otherwise             

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge - 3 a \bmod \pi = 0 \\\sin{\left(3 a \right)} & \text{otherwise} \end{cases}$$

           2            
------------------------
/        1    \    /3*a\
|1 + ---------|*tan|---|
|       2/3*a\|    \ 2 /
|    tan |---||         
\        \ 2 //         

$$\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{3 a}{2} \right)}}\right) \tan{\left(\frac{3 a}{2} \right)}}$$

   /      pi\
cos|3*a - --|
   \      2 /

$$\cos{\left(3 a - \frac{\pi}{2} \right)}$$

/                      0                        for And(im(a) = 0, -3*a mod pi = 0)
|                                                                                  

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge - 3 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\sin{\left(3 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}$$

/           0              for And(im(a) = 0, -3*a mod pi = 0)
|                                                             
|           2                                                 
|------------------------               otherwise             

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge - 3 a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{3 a}{2} \right)}}\right) \cot{\left(\frac{3 a}{2} \right)}} & \text{otherwise} \end{cases}$$

/      0        for And(im(a) = 0, -3*a mod pi = 0)
|                                                  
|      1                                           
<-------------               otherwise             
|   /      pi\                                     
|sec|3*a - --|                                     
\   \      2 /                                     

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge - 3 a \bmod \pi = 0 \\\frac{1}{\sec{\left(3 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}$$

/   0      for And(im(a) = 0, -3*a mod pi = 0)
|                                             
<   1                                         
|--------               otherwise             
\csc(3*a)                                     

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge - 3 a \bmod \pi = 0 \\\frac{1}{\csc{\left(3 a \right)}} & \text{otherwise} \end{cases}$$

       /3*a\ 
  2*cot|---| 
       \ 2 / 
-------------
       2/3*a\
1 + cot |---|
        \ 2 /

$$\frac{2 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1}$$

/   0      for And(im(a) = 0, 3*a mod pi = 0)
<                                            
\sin(3*a)              otherwise             

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\sin{\left(3 a \right)} & \text{otherwise} \end{cases}$$

   /pi      \
cos|-- - 3*a|
   \2       /

$$\cos{\left(- 3 a + \frac{\pi}{2} \right)}$$

Раскрыть выражение

       3              
- 4*sin (a) + 3*sin(a)

$$- 4 \sin^{3}{\left(a \right)} + 3 \sin{\left(a \right)}$$