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sin(270-a)еслиa=1/2 (упростите выражение)

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Примеры

Решение

Вы ввели [src]
sin(270 - a)
$$\sin{\left(270 - a \right)}$$
Подстановка условия [src]
sin(270 - a) при a = 1/2
подставляем
sin(270 - a)
$$\sin{\left(270 - a \right)}$$
-sin(-270 + a)
$$- \sin{\left(a - 270 \right)}$$
переменные
a = 1/2
$$a = \frac{1}{2}$$
-sin(-270 + (1/2))
$$- \sin{\left((1/2) - 270 \right)}$$
-sin(-270 + 1/2)
$$- \sin{\left(-270 + \frac{1}{2} \right)}$$
sin(539/2)
$$\sin{\left(\frac{539}{2} \right)}$$
Степени [src]
-sin(-270 + a)
$$- \sin{\left(a - 270 \right)}$$
   /   I*(-270 + a)    I*(270 - a)\ 
-I*\- e             + e           / 
------------------------------------
                 2
$$- \frac{i \left(e^{i \left(270 - a\right)} - e^{i \left(a - 270\right)}\right)}{2}$$
Численный ответ [src]
sin(270 - a)
Рациональный знаменатель [src]
-sin(-270 + a)
$$- \sin{\left(a - 270 \right)}$$
Объединение рациональных выражений [src]
-sin(-270 + a)
$$- \sin{\left(a - 270 \right)}$$
Общее упрощение [src]
-sin(-270 + a)
$$- \sin{\left(a - 270 \right)}$$
Собрать выражение [src]
-sin(-270 + a)
$$- \sin{\left (a - 270 \right )}$$
Общий знаменатель [src]
-sin(-270 + a)
$$- \sin{\left(a - 270 \right)}$$
Тригонометрическая часть [src]
               -2                 
----------------------------------
/          1       \    /       a\
|1 + --------------|*cot|-135 + -|
|       2/       a\|    \       2/
|    cot |-135 + -||              
\        \       2//
$$- \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} - 135 \right)}}\right) \cot{\left(\frac{a}{2} - 135 \right)}}$$
       /       a\ 
 -2*tan|-135 + -| 
       \       2/ 
------------------
       2/       a\
1 + tan |-135 + -|
        \       2/
$$- \frac{2 \tan{\left(\frac{a}{2} - 135 \right)}}{\tan^{2}{\left(\frac{a}{2} - 135 \right)} + 1}$$
/        0           for And(im(a) = 0, (270 - a - 85*pi) mod pi = 0)
|                                                                    
|       /       a\                                                   
| -2*tan|-135 + -|                                                   
<       \       2/                                                   
|------------------                     otherwise                    
|       2/       a\                                                  
|1 + tan |-135 + -|                                                  
\        \       2/
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(- a - 85 \pi + 270\right) \bmod \pi = 0 \\- \frac{2 \tan{\left(\frac{a}{2} - 135 \right)}}{\tan^{2}{\left(\frac{a}{2} - 135 \right)} + 1} & \text{otherwise} \end{cases}$$
-sin(-270 + a)
$$- \sin{\left(a - 270 \right)}$$
        1        
-----------------
   /          pi\
sec|270 - a - --|
   \          2 /
$$\frac{1}{\sec{\left(- a - \frac{\pi}{2} + 270 \right)}}$$
/        0           for And(im(a) = 0, (270 - a - 85*pi) mod pi = 0)
|                                                                    
|       -1                                                           
<------------------                     otherwise                    
|   /           pi\                                                  
|sec|-270 + a - --|                                                  
\   \           2 /
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(- a - 85 \pi + 270\right) \bmod \pi = 0 \\- \frac{1}{\sec{\left(a - 270 - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}$$
               /       a   pi\         
         -2*cos|-135 + - - --|         
               \       2   2 /         
---------------------------------------
/       2/       a   pi\\              
|    cos |-135 + - - --||              
|        \       2   2 /|    /       a\
|1 + -------------------|*cos|-135 + -|
|          2/       a\  |    \       2/
|       cos |-135 + -|  |              
\           \       2/  /
$$- \frac{2 \cos{\left(\frac{a}{2} - 135 - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - 135 - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - 135 \right)}}\right) \cos{\left(\frac{a}{2} - 135 \right)}}$$
/      0         for And(im(a) = 0, (270 - a - 85*pi) mod pi = 0)
<                                                                
\-sin(-270 + a)                     otherwise
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(- a - 85 \pi + 270\right) \bmod \pi = 0 \\- \sin{\left(a - 270 \right)} & \text{otherwise} \end{cases}$$
       -1         
------------------
   /           pi\
sec|-270 + a - --|
   \           2 /
$$- \frac{1}{\sec{\left(a - 270 - \frac{\pi}{2} \right)}}$$
     /       a\                    /       a\
- tan|-135 + -| - cos(-270 + a)*tan|-135 + -|
     \       2/                    \       2/
$$- \cos{\left(a - 270 \right)} \tan{\left(\frac{a}{2} - 135 \right)} - \tan{\left(\frac{a}{2} - 135 \right)}$$
    /           pi\
-cos|-270 + a - --|
    \           2 /
$$- \cos{\left(a - 270 - \frac{\pi}{2} \right)}$$
/      0        for And(im(a) = 0, (270 - a - 85*pi) mod pi = 0)
|                                                               
<     -1                                                        
|-------------                     otherwise                    
\csc(-270 + a)
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(- a - 85 \pi + 270\right) \bmod \pi = 0 \\- \frac{1}{\csc{\left(a - 270 \right)}} & \text{otherwise} \end{cases}$$
/         0           for And(im(a) = 0, (270 - a - 85*pi) mod pi = 0)
|                                                                     
<    /           pi\                                                  
|-cos|-270 + a - --|                     otherwise                    
\    \           2 /
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(- a - 85 \pi + 270\right) \bmod \pi = 0 \\- \cos{\left(a - 270 - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}$$
                    /       a\              
              -2*sec|-135 + -|              
                    \       2/              
--------------------------------------------
/          2/       a\  \                   
|       sec |-135 + -|  |                   
|           \       2/  |    /       a   pi\
|1 + -------------------|*sec|-135 + - - --|
|       2/       a   pi\|    \       2   2 /
|    sec |-135 + - - --||                   
\        \       2   2 //
$$- \frac{2 \sec{\left(\frac{a}{2} - 135 \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} - 135 \right)}}{\sec^{2}{\left(\frac{a}{2} - 135 - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - 135 - \frac{\pi}{2} \right)}}$$
/        0           for And(im(a) = 0, (270 - a - 85*pi) mod pi = 0)
|                                                                    
|       /       a\                                                   
| -2*cot|-135 + -|                                                   
<       \       2/                                                   
|------------------                     otherwise                    
|       2/       a\                                                  
|1 + cot |-135 + -|                                                  
\        \       2/
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(- a - 85 \pi + 270\right) \bmod \pi = 0 \\- \frac{2 \cot{\left(\frac{a}{2} - 135 \right)}}{\cot^{2}{\left(\frac{a}{2} - 135 \right)} + 1} & \text{otherwise} \end{cases}$$
      /       a\    /       a\
-2*sec|-135 + -|*sin|-135 + -|
      \       2/    \       2/
------------------------------
             2/       a\      
      1 + tan |-135 + -|      
              \       2/
$$- \frac{2 \sin{\left(\frac{a}{2} - 135 \right)} \sec{\left(\frac{a}{2} - 135 \right)}}{\tan^{2}{\left(\frac{a}{2} - 135 \right)} + 1}$$
 //        0           for And(im(a) = 0, (-270 + a + 86*pi) mod pi = 0)\
 ||                                                                     |
 ||      /       a\                                                     |
 || 2*cot|-135 + -|                                                     |
-|<      \       2/                                                     |
 ||------------------                      otherwise                    |
 ||       2/       a\                                                   |
 ||1 + cot |-135 + -|                                                   |
 \\        \       2/                                                   /
$$- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(a - 270 + 86 \pi\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} - 135 \right)}}{\cot^{2}{\left(\frac{a}{2} - 135 \right)} + 1} & \text{otherwise} \end{cases}$$
     -1      
-------------
csc(-270 + a)
$$- \frac{1}{\csc{\left(a - 270 \right)}}$$
 //      0        for And(im(a) = 0, (-270 + a + 86*pi) mod pi = 0)\
-|<                                                                |
 \\sin(-270 + a)                      otherwise                    /
$$- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(a - 270 + 86 \pi\right) \bmod \pi = 0 \\\sin{\left(a - 270 \right)} & \text{otherwise} \end{cases}$$
/                                    0                                      for And(im(a) = 0, (270 - a - 85*pi) mod pi = 0)
|                                                                                                                           
| //        0           for And(im(a) = 0, (-270 + a + 86*pi) mod pi = 0)\                                                  
| ||                                                                     |                                                  
| ||      /       a\                                                     |                                                  
< || 2*cot|-135 + -|                                                     |                                                  
|-|<      \       2/                                                     |                     otherwise                    
| ||------------------                      otherwise                    |                                                  
| ||       2/       a\                                                   |                                                  
| ||1 + cot |-135 + -|                                                   |                                                  
\ \\        \       2/                                                   /
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(- a - 85 \pi + 270\right) \bmod \pi = 0 \\- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(a - 270 + 86 \pi\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} - 135 \right)}}{\cot^{2}{\left(\frac{a}{2} - 135 \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}$$
               2/       a\          
         -4*sin |-135 + -|          
                \       2/          
------------------------------------
/         4/       a\\              
|    4*sin |-135 + -||              
|          \       2/|              
|1 + ----------------|*sin(-270 + a)
|        2           |              
\     sin (-270 + a) /
$$- \frac{4 \sin^{2}{\left(\frac{a}{2} - 135 \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} - 135 \right)}}{\sin^{2}{\left(a - 270 \right)}} + 1\right) \sin{\left(a - 270 \right)}}$$
   /          pi\
cos|270 - a - --|
   \          2 /
$$\cos{\left(- a - \frac{\pi}{2} + 270 \right)}$$
/                                 0                                    for And(im(a) = 0, (270 - a - 85*pi) mod pi = 0)
|                                                                                                                      
< //      0        for And(im(a) = 0, (-270 + a + 86*pi) mod pi = 0)\                                                  
|-|<                                                                |                     otherwise                    
\ \\sin(-270 + a)                      otherwise                    /
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(- a - 85 \pi + 270\right) \bmod \pi = 0 \\- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(a - 270 + 86 \pi\right) \bmod \pi = 0 \\\sin{\left(a - 270 \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}$$
               /      pi   a\         
         -2*csc|135 + -- - -|         
               \      2    2/         
--------------------------------------
/       2/      pi   a\\              
|    csc |135 + -- - -||              
|        \      2    2/|    /       a\
|1 + ------------------|*csc|-135 + -|
|         2/       a\  |    \       2/
|      csc |-135 + -|  |              
\          \       2/  /
$$- \frac{2 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} + 135 \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} + 135 \right)}}{\csc^{2}{\left(\frac{a}{2} - 135 \right)}}\right) \csc{\left(\frac{a}{2} - 135 \right)}}$$
Комбинаторика [src]
-sin(-270 + a)
$$- \sin{\left(a - 270 \right)}$$
Раскрыть выражение [src]
cos(a)*sin(270) - cos(270)*sin(a)
$$- \sin{\left(a \right)} \cos{\left(270 \right)} + \sin{\left(270 \right)} \cos{\left(a \right)}$$