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sin(270-x)еслиx=-1/2 (упростите выражение)

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Примеры

Решение

Вы ввели [src]
sin(270 - x)
sin⁡(270−x)\sin{\left(270 - x \right)}
Подстановка условия [src]
sin(270 - x) при x = -1/2
подставляем
sin(270 - x)
sin⁡(270−x)\sin{\left(270 - x \right)}
-sin(-270 + x)
−sin⁡(x−270)- \sin{\left(x - 270 \right)}
переменные
x = -1/2
x=−12x = - \frac{1}{2}
-sin(-270 + (-1/2))
−sin⁡((−1/2)−270)- \sin{\left((-1/2) - 270 \right)}
-sin(-270 - 1/2)
−sin⁡(−270−12)- \sin{\left(-270 - \frac{1}{2} \right)}
sin(541/2)
sin⁡(5412)\sin{\left(\frac{541}{2} \right)}
Степени [src]
   /   I*(-270 + x)    I*(270 - x)\ 
-I*\- e             + e           / 
------------------------------------
                 2
−i(ei(270−x)−ei(x−270))2- \frac{i \left(e^{i \left(270 - x\right)} - e^{i \left(x - 270\right)}\right)}{2}
-sin(-270 + x)
−sin⁡(x−270)- \sin{\left(x - 270 \right)}
Численный ответ [src]
sin(270 - x)
Рациональный знаменатель [src]
-sin(-270 + x)
−sin⁡(x−270)- \sin{\left(x - 270 \right)}
Объединение рациональных выражений [src]
-sin(-270 + x)
−sin⁡(x−270)- \sin{\left(x - 270 \right)}
Общее упрощение [src]
-sin(-270 + x)
−sin⁡(x−270)- \sin{\left(x - 270 \right)}
Собрать выражение [src]
-sin(-270 + x)
−sin⁡(x−270)- \sin{\left (x - 270 \right )}
Комбинаторика [src]
-sin(-270 + x)
−sin⁡(x−270)- \sin{\left(x - 270 \right)}
Общий знаменатель [src]
-sin(-270 + x)
−sin⁡(x−270)- \sin{\left(x - 270 \right)}
Тригонометрическая часть [src]
               2/       x\          
         -4*sin |-135 + -|          
                \       2/          
------------------------------------
/         4/       x\\              
|    4*sin |-135 + -||              
|          \       2/|              
|1 + ----------------|*sin(-270 + x)
|        2           |              
\     sin (-270 + x) /
−4sin⁡2(x2−135)(4sin⁡4(x2−135)sin⁡2(x−270)+1)sin⁡(x−270)- \frac{4 \sin^{2}{\left(\frac{x}{2} - 135 \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} - 135 \right)}}{\sin^{2}{\left(x - 270 \right)}} + 1\right) \sin{\left(x - 270 \right)}}
 //        0           for And(im(x) = 0, (-270 + x + 86*pi) mod pi = 0)\
 ||                                                                     |
 ||      /       x\                                                     |
 || 2*cot|-135 + -|                                                     |
-|<      \       2/                                                     |
 ||------------------                      otherwise                    |
 ||       2/       x\                                                   |
 ||1 + cot |-135 + -|                                                   |
 \\        \       2/                                                   /
−{0for im⁡(x)=0∧(x−270+86π) mod π=02cot⁡(x2−135)cot⁡2(x2−135)+1otherwise- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(x - 270 + 86 \pi\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} - 135 \right)}}{\cot^{2}{\left(\frac{x}{2} - 135 \right)} + 1} & \text{otherwise} \end{cases}
                    /       x\              
              -2*sec|-135 + -|              
                    \       2/              
--------------------------------------------
/          2/       x\  \                   
|       sec |-135 + -|  |                   
|           \       2/  |    /       x   pi\
|1 + -------------------|*sec|-135 + - - --|
|       2/       x   pi\|    \       2   2 /
|    sec |-135 + - - --||                   
\        \       2   2 //
−2sec⁡(x2−135)(sec⁡2(x2−135)sec⁡2(x2−135−π2)+1)sec⁡(x2−135−π2)- \frac{2 \sec{\left(\frac{x}{2} - 135 \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} - 135 \right)}}{\sec^{2}{\left(\frac{x}{2} - 135 - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} - 135 - \frac{\pi}{2} \right)}}
       /       x\ 
 -2*tan|-135 + -| 
       \       2/ 
------------------
       2/       x\
1 + tan |-135 + -|
        \       2/
−2tan⁡(x2−135)tan⁡2(x2−135)+1- \frac{2 \tan{\left(\frac{x}{2} - 135 \right)}}{\tan^{2}{\left(\frac{x}{2} - 135 \right)} + 1}
/         0           for And(im(x) = 0, (270 - x - 85*pi) mod pi = 0)
|                                                                     
<    /           pi\                                                  
|-cos|-270 + x - --|                     otherwise                    
\    \           2 /
{0for im⁡(x)=0∧(−x−85π+270) mod π=0−cos⁡(x−270−π2)otherwise\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(- x - 85 \pi + 270\right) \bmod \pi = 0 \\- \cos{\left(x - 270 - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}
    /           pi\
-cos|-270 + x - --|
    \           2 /
−cos⁡(x−270−π2)- \cos{\left(x - 270 - \frac{\pi}{2} \right)}
/      0         for And(im(x) = 0, (270 - x - 85*pi) mod pi = 0)
<                                                                
\-sin(-270 + x)                     otherwise
{0for im⁡(x)=0∧(−x−85π+270) mod π=0−sin⁡(x−270)otherwise\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(- x - 85 \pi + 270\right) \bmod \pi = 0 \\- \sin{\left(x - 270 \right)} & \text{otherwise} \end{cases}
 //      0        for And(im(x) = 0, (-270 + x + 86*pi) mod pi = 0)\
-|<                                                                |
 \\sin(-270 + x)                      otherwise                    /
−{0for im⁡(x)=0∧(x−270+86π) mod π=0sin⁡(x−270)otherwise- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(x - 270 + 86 \pi\right) \bmod \pi = 0 \\\sin{\left(x - 270 \right)} & \text{otherwise} \end{cases}
/        0           for And(im(x) = 0, (270 - x - 85*pi) mod pi = 0)
|                                                                    
|       /       x\                                                   
| -2*cot|-135 + -|                                                   
<       \       2/                                                   
|------------------                     otherwise                    
|       2/       x\                                                  
|1 + cot |-135 + -|                                                  
\        \       2/
{0for im⁡(x)=0∧(−x−85π+270) mod π=0−2cot⁡(x2−135)cot⁡2(x2−135)+1otherwise\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(- x - 85 \pi + 270\right) \bmod \pi = 0 \\- \frac{2 \cot{\left(\frac{x}{2} - 135 \right)}}{\cot^{2}{\left(\frac{x}{2} - 135 \right)} + 1} & \text{otherwise} \end{cases}
     /       x\                    /       x\
- tan|-135 + -| - cos(-270 + x)*tan|-135 + -|
     \       2/                    \       2/
−cos⁡(x−270)tan⁡(x2−135)−tan⁡(x2−135)- \cos{\left(x - 270 \right)} \tan{\left(\frac{x}{2} - 135 \right)} - \tan{\left(\frac{x}{2} - 135 \right)}
-sin(-270 + x)
−sin⁡(x−270)- \sin{\left(x - 270 \right)}
               /      pi   x\         
         -2*csc|135 + -- - -|         
               \      2    2/         
--------------------------------------
/       2/      pi   x\\              
|    csc |135 + -- - -||              
|        \      2    2/|    /       x\
|1 + ------------------|*csc|-135 + -|
|         2/       x\  |    \       2/
|      csc |-135 + -|  |              
\          \       2/  /
−2csc⁡(−x2+π2+135)(1+csc⁡2(−x2+π2+135)csc⁡2(x2−135))csc⁡(x2−135)- \frac{2 \csc{\left(- \frac{x}{2} + \frac{\pi}{2} + 135 \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} + 135 \right)}}{\csc^{2}{\left(\frac{x}{2} - 135 \right)}}\right) \csc{\left(\frac{x}{2} - 135 \right)}}
        1        
-----------------
   /          pi\
sec|270 - x - --|
   \          2 /
1sec⁡(−x−π2+270)\frac{1}{\sec{\left(- x - \frac{\pi}{2} + 270 \right)}}
     -1      
-------------
csc(-270 + x)
−1csc⁡(x−270)- \frac{1}{\csc{\left(x - 270 \right)}}
/        0           for And(im(x) = 0, (270 - x - 85*pi) mod pi = 0)
|                                                                    
|       -1                                                           
<------------------                     otherwise                    
|   /           pi\                                                  
|sec|-270 + x - --|                                                  
\   \           2 /
{0for im⁡(x)=0∧(−x−85π+270) mod π=0−1sec⁡(x−270−π2)otherwise\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(- x - 85 \pi + 270\right) \bmod \pi = 0 \\- \frac{1}{\sec{\left(x - 270 - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}
               /       x   pi\         
         -2*cos|-135 + - - --|         
               \       2   2 /         
---------------------------------------
/       2/       x   pi\\              
|    cos |-135 + - - --||              
|        \       2   2 /|    /       x\
|1 + -------------------|*cos|-135 + -|
|          2/       x\  |    \       2/
|       cos |-135 + -|  |              
\           \       2/  /
−2cos⁡(x2−135−π2)(1+cos⁡2(x2−135−π2)cos⁡2(x2−135))cos⁡(x2−135)- \frac{2 \cos{\left(\frac{x}{2} - 135 - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - 135 - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - 135 \right)}}\right) \cos{\left(\frac{x}{2} - 135 \right)}}
               -2                 
----------------------------------
/          1       \    /       x\
|1 + --------------|*cot|-135 + -|
|       2/       x\|    \       2/
|    cot |-135 + -||              
\        \       2//
−2(1+1cot⁡2(x2−135))cot⁡(x2−135)- \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} - 135 \right)}}\right) \cot{\left(\frac{x}{2} - 135 \right)}}
      /       x\    /       x\
-2*sec|-135 + -|*sin|-135 + -|
      \       2/    \       2/
------------------------------
             2/       x\      
      1 + tan |-135 + -|      
              \       2/
−2sin⁡(x2−135)sec⁡(x2−135)tan⁡2(x2−135)+1- \frac{2 \sin{\left(\frac{x}{2} - 135 \right)} \sec{\left(\frac{x}{2} - 135 \right)}}{\tan^{2}{\left(\frac{x}{2} - 135 \right)} + 1}
/                                 0                                    for And(im(x) = 0, (270 - x - 85*pi) mod pi = 0)
|                                                                                                                      
< //      0        for And(im(x) = 0, (-270 + x + 86*pi) mod pi = 0)\                                                  
|-|<                                                                |                     otherwise                    
\ \\sin(-270 + x)                      otherwise                    /
{0for im⁡(x)=0∧(−x−85π+270) mod π=0−{0for im⁡(x)=0∧(x−270+86π) mod π=0sin⁡(x−270)otherwiseotherwise\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(- x - 85 \pi + 270\right) \bmod \pi = 0 \\- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(x - 270 + 86 \pi\right) \bmod \pi = 0 \\\sin{\left(x - 270 \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}
   /          pi\
cos|270 - x - --|
   \          2 /
cos⁡(−x−π2+270)\cos{\left(- x - \frac{\pi}{2} + 270 \right)}
/      0        for And(im(x) = 0, (270 - x - 85*pi) mod pi = 0)
|                                                               
<     -1                                                        
|-------------                     otherwise                    
\csc(-270 + x)
{0for im⁡(x)=0∧(−x−85π+270) mod π=0−1csc⁡(x−270)otherwise\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(- x - 85 \pi + 270\right) \bmod \pi = 0 \\- \frac{1}{\csc{\left(x - 270 \right)}} & \text{otherwise} \end{cases}
/        0           for And(im(x) = 0, (270 - x - 85*pi) mod pi = 0)
|                                                                    
|       /       x\                                                   
| -2*tan|-135 + -|                                                   
<       \       2/                                                   
|------------------                     otherwise                    
|       2/       x\                                                  
|1 + tan |-135 + -|                                                  
\        \       2/
{0for im⁡(x)=0∧(−x−85π+270) mod π=0−2tan⁡(x2−135)tan⁡2(x2−135)+1otherwise\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(- x - 85 \pi + 270\right) \bmod \pi = 0 \\- \frac{2 \tan{\left(\frac{x}{2} - 135 \right)}}{\tan^{2}{\left(\frac{x}{2} - 135 \right)} + 1} & \text{otherwise} \end{cases}
       -1         
------------------
   /           pi\
sec|-270 + x - --|
   \           2 /
−1sec⁡(x−270−π2)- \frac{1}{\sec{\left(x - 270 - \frac{\pi}{2} \right)}}
/                                    0                                      for And(im(x) = 0, (270 - x - 85*pi) mod pi = 0)
|                                                                                                                           
| //        0           for And(im(x) = 0, (-270 + x + 86*pi) mod pi = 0)\                                                  
| ||                                                                     |                                                  
| ||      /       x\                                                     |                                                  
< || 2*cot|-135 + -|                                                     |                                                  
|-|<      \       2/                                                     |                     otherwise                    
| ||------------------                      otherwise                    |                                                  
| ||       2/       x\                                                   |                                                  
| ||1 + cot |-135 + -|                                                   |                                                  
\ \\        \       2/                                                   /
{0for im⁡(x)=0∧(−x−85π+270) mod π=0−{0for im⁡(x)=0∧(x−270+86π) mod π=02cot⁡(x2−135)cot⁡2(x2−135)+1otherwiseotherwise\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(- x - 85 \pi + 270\right) \bmod \pi = 0 \\- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(x - 270 + 86 \pi\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} - 135 \right)}}{\cot^{2}{\left(\frac{x}{2} - 135 \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}
Раскрыть выражение [src]
cos(x)*sin(270) - cos(270)*sin(x)
−sin⁡(x)cos⁡(270)+sin⁡(270)cos⁡(x)- \sin{\left(x \right)} \cos{\left(270 \right)} + \sin{\left(270 \right)} \cos{\left(x \right)}