Господин Экзамен

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Упрощение выражений/ cos(90+t) если t=2

cos(90+t) если t=2

Выражение, которое надо упростить:

Решение

Вы ввели [src] 
cos(90 + t)
cos⁡(t+90)\cos{\left(t + 90 \right)} 
cos(90 + t)
Подстановка условия [src] 
cos(90 + t) при t = 2
подставляем
cos(90 + t)
cos⁡(t+90)\cos{\left(t + 90 \right)} 
cos(90 + t)
cos⁡(t+90)\cos{\left(t + 90 \right)} 
переменные
t = 2
t=2t = 2 
cos(90 + (2))
cos⁡((2)+90)\cos{\left((2) + 90 \right)} 
cos(90 + 2)
cos⁡(2+90)\cos{\left(2 + 90 \right)} 
cos(92)
cos⁡(92)\cos{\left(92 \right)} 
cos(92)
Раскрыть выражение [src] 
cos(90)*cos(t) - sin(90)*sin(t)
−sin⁡(90)sin⁡(t)+cos⁡(90)cos⁡(t)- \sin{\left(90 \right)} \sin{\left(t \right)} + \cos{\left(90 \right)} \cos{\left(t \right)} 
cos(90)*cos(t) - sin(90)*sin(t)
Тригонометрическая часть [src] 
     1     
-----------
sec(90 + t)
1sec⁡(t+90)\frac{1}{\sec{\left(t + 90 \right)}} 
   /         pi\
sin|90 + t + --|
   \         2 /
sin⁡(t+π2+90)\sin{\left(t + \frac{\pi}{2} + 90 \right)} 
        1        
-----------------
   /      pi    \
csc|-90 + -- - t|
   \      2     /
1csc⁡(−t−90+π2)\frac{1}{\csc{\left(- t - 90 + \frac{\pi}{2} \right)}} 
        2/     t\
-1 + cot |45 + -|
         \     2/
-----------------
        2/     t\
 1 + cot |45 + -|
         \     2/
cot⁡2(t2+45)−1cot⁡2(t2+45)+1\frac{\cot^{2}{\left(\frac{t}{2} + 45 \right)} - 1}{\cot^{2}{\left(\frac{t}{2} + 45 \right)} + 1} 
       2/     t\
1 - tan |45 + -|
        \     2/
----------------
       2/     t\
1 + tan |45 + -|
        \     2/
−tan⁡2(t2+45)+1tan⁡2(t2+45)+1\frac{- \tan^{2}{\left(\frac{t}{2} + 45 \right)} + 1}{\tan^{2}{\left(\frac{t}{2} + 45 \right)} + 1} 
         1      
1 - ------------
       2/     t\
    cot |45 + -|
        \     2/
----------------
         1      
1 + ------------
       2/     t\
    cot |45 + -|
        \     2/
1−1cot⁡2(t2+45)1+1cot⁡2(t2+45)\frac{1 - \frac{1}{\cot^{2}{\left(\frac{t}{2} + 45 \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} + 45 \right)}}} 
       /     t   pi\ 
  2*tan|45 + - + --| 
       \     2   4 / 
---------------------
       2/     t   pi\
1 + tan |45 + - + --|
        \     2   4 /
2tan⁡(t2+π4+45)tan⁡2(t2+π4+45)+1\frac{2 \tan{\left(\frac{t}{2} + \frac{\pi}{4} + 45 \right)}}{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} + 45 \right)} + 1} 
/     1       for (90 + t - 28*pi) mod 2*pi = 0
<                                              
\cos(90 + t)              otherwise
{1for (t−28π+90) mod 2π=0cos⁡(t+90)otherwise\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\cos{\left(t + 90 \right)} & \text{otherwise} \end{cases} 
/     1       for (90 + t - 28*pi) mod 2*pi = 0
|                                              
<     1                                        
|-----------              otherwise            
\sec(90 + t)
{1for (t−28π+90) mod 2π=01sec⁡(t+90)otherwise\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(t + 90 \right)}} & \text{otherwise} \end{cases} 
/       1          for (90 + t - 28*pi) mod 2*pi = 0
|                                                   
<   /         pi\                                   
|sin|90 + t + --|              otherwise            
\   \         2 /
{1for (t−28π+90) mod 2π=0sin⁡(t+π2+90)otherwise\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\sin{\left(t + \frac{\pi}{2} + 90 \right)} & \text{otherwise} \end{cases} 
/        1          for (90 + t - 28*pi) mod 2*pi = 0
|                                                    
|        1                                           
<-----------------              otherwise            
|   /      pi    \                                   
|csc|-90 + -- - t|                                   
\   \      2     /
{1for (t−28π+90) mod 2π=01csc⁡(−t−90+π2)otherwise\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- t - 90 + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases} 
         4/     t\
    4*sin |45 + -|
          \     2/
1 - --------------
        2         
     sin (90 + t) 
------------------
         4/     t\
    4*sin |45 + -|
          \     2/
1 + --------------
        2         
     sin (90 + t)
−4sin⁡4(t2+45)sin⁡2(t+90)+14sin⁡4(t2+45)sin⁡2(t+90)+1\frac{- \frac{4 \sin^{4}{\left(\frac{t}{2} + 45 \right)}}{\sin^{2}{\left(t + 90 \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{t}{2} + 45 \right)}}{\sin^{2}{\left(t + 90 \right)}} + 1} 
/        1          for (90 + t - 28*pi) mod 2*pi = 0
|                                                    
|        2/     t\                                   
|-1 + cot |45 + -|                                   
<         \     2/                                   
|-----------------              otherwise            
|        2/     t\                                   
| 1 + cot |45 + -|                                   
\         \     2/
{1for (t−28π+90) mod 2π=0cot⁡2(t2+45)−1cot⁡2(t2+45)+1otherwise\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} + 45 \right)} - 1}{\cot^{2}{\left(\frac{t}{2} + 45 \right)} + 1} & \text{otherwise} \end{cases} 
/       1          for (90 + t - 28*pi) mod 2*pi = 0
|                                                   
|       2/     t\                                   
|1 - tan |45 + -|                                   
<        \     2/                                   
|----------------              otherwise            
|       2/     t\                                   
|1 + tan |45 + -|                                   
\        \     2/
{1for (t−28π+90) mod 2π=0−tan⁡2(t2+45)+1tan⁡2(t2+45)+1otherwise\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{t}{2} + 45 \right)} + 1}{\tan^{2}{\left(\frac{t}{2} + 45 \right)} + 1} & \text{otherwise} \end{cases} 
/                                        /         55*pi\           
|                0                   for |90 + t - -----| mod pi = 0
|                                        \           2  /           
<                                                                   
|                     /     t   pi\                                 
|(1 + sin(90 + t))*cot|45 + - + --|             otherwise           
\                     \     2   4 /
{0for (t−55π2+90) mod π=0(sin⁡(t+90)+1)cot⁡(t2+π4+45)otherwise\begin{cases} 0 & \text{for}\: \left(t - \frac{55 \pi}{2} + 90\right) \bmod \pi = 0 \\\left(\sin{\left(t + 90 \right)} + 1\right) \cot{\left(\frac{t}{2} + \frac{\pi}{4} + 45 \right)} & \text{otherwise} \end{cases} 
       2/     t   pi\
    cos |45 + - - --|
        \     2   2 /
1 - -----------------
          2/     t\  
       cos |45 + -|  
           \     2/  
---------------------
       2/     t   pi\
    cos |45 + - - --|
        \     2   2 /
1 + -----------------
          2/     t\  
       cos |45 + -|  
           \     2/
1−cos⁡2(t2−π2+45)cos⁡2(t2+45)1+cos⁡2(t2−π2+45)cos⁡2(t2+45)\frac{1 - \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} + 45 \right)}}{\cos^{2}{\left(\frac{t}{2} + 45 \right)}}}{1 + \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} + 45 \right)}}{\cos^{2}{\left(\frac{t}{2} + 45 \right)}}} 
/        1          for (90 + t - 28*pi) mod 2*pi = 0
|                                                    
|          1                                         
|-1 + ------------                                   
|        2/     t\                                   
|     tan |45 + -|                                   
<         \     2/                                   
|-----------------              otherwise            
|          1                                         
| 1 + ------------                                   
|        2/     t\                                   
|     tan |45 + -|                                   
\         \     2/
{1for (t−28π+90) mod 2π=0−1+1tan⁡2(t2+45)1+1tan⁡2(t2+45)otherwise\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} + 45 \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} + 45 \right)}}} & \text{otherwise} \end{cases} 
          2/     t\  
       sec |45 + -|  
           \     2/  
1 - -----------------
       2/     t   pi\
    sec |45 + - - --|
        \     2   2 /
---------------------
          2/     t\  
       sec |45 + -|  
           \     2/  
1 + -----------------
       2/     t   pi\
    sec |45 + - - --|
        \     2   2 /
−sec⁡2(t2+45)sec⁡2(t2−π2+45)+1sec⁡2(t2+45)sec⁡2(t2−π2+45)+1\frac{- \frac{\sec^{2}{\left(\frac{t}{2} + 45 \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} + 45 \right)}} + 1}{\frac{\sec^{2}{\left(\frac{t}{2} + 45 \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} + 45 \right)}} + 1} 
       2/      pi   t\
    csc |-45 + -- - -|
        \      2    2/
1 - ------------------
          2/     t\   
       csc |45 + -|   
           \     2/   
----------------------
       2/      pi   t\
    csc |-45 + -- - -|
        \      2    2/
1 + ------------------
          2/     t\   
       csc |45 + -|   
           \     2/
1−csc⁡2(−t2−45+π2)csc⁡2(t2+45)1+csc⁡2(−t2−45+π2)csc⁡2(t2+45)\frac{1 - \frac{\csc^{2}{\left(- \frac{t}{2} - 45 + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} + 45 \right)}}}{1 + \frac{\csc^{2}{\left(- \frac{t}{2} - 45 + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} + 45 \right)}}} 
/                           /         55*pi\           
|          0            for |90 + t - -----| mod pi = 0
|                           \           2  /           
|                                                      
|       /     t   pi\                                  
<  2*cot|45 + - + --|                                  
|       \     2   4 /                                  
|---------------------             otherwise           
|       2/     t   pi\                                 
|1 + cot |45 + - + --|                                 
\        \     2   4 /
{0for (t−55π2+90) mod π=02cot⁡(t2+π4+45)cot⁡2(t2+π4+45)+1otherwise\begin{cases} 0 & \text{for}\: \left(t - \frac{55 \pi}{2} + 90\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{t}{2} + \frac{\pi}{4} + 45 \right)}}{\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} + 45 \right)} + 1} & \text{otherwise} \end{cases} 
/                       1                         for (90 + t - 28*pi) mod 2*pi = 0
|
{1for (t−28π+90) mod 2π=0{1for (t−28π+90) mod 2π=0cos⁡(t+90)otherwiseotherwise\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\cos{\left(t + 90 \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases} 
/              1                for (90 + t - 28*pi) mod 2*pi = 0
|                                                                
|   2                4/     t\                                   
|sin (90 + t) - 4*sin |45 + -|                                   
<                     \     2/                                   
|-----------------------------              otherwise            
|   2                4/     t\                                   
|sin (90 + t) + 4*sin |45 + -|                                   
\                     \     2/
{1for (t−28π+90) mod 2π=0−4sin⁡4(t2+45)+sin⁡2(t+90)4sin⁡4(t2+45)+sin⁡2(t+90)otherwise\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{- 4 \sin^{4}{\left(\frac{t}{2} + 45 \right)} + \sin^{2}{\left(t + 90 \right)}}{4 \sin^{4}{\left(\frac{t}{2} + 45 \right)} + \sin^{2}{\left(t + 90 \right)}} & \text{otherwise} \end{cases} 
/         1           for (90 + t - 28*pi) mod 2*pi = 0
|                                                      
|         2                                            
|      sin (90 + t)                                    
|-1 + --------------                                   
|          4/     t\                                   
|     4*sin |45 + -|                                   
<           \     2/                                   
|-------------------              otherwise            
|         2                                            
|      sin (90 + t)                                    
| 1 + --------------                                   
|          4/     t\                                   
|     4*sin |45 + -|                                   
\           \     2/
{1for (t−28π+90) mod 2π=0−1+sin⁡2(t+90)4sin⁡4(t2+45)1+sin⁡2(t+90)4sin⁡4(t2+45)otherwise\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(t + 90 \right)}}{4 \sin^{4}{\left(\frac{t}{2} + 45 \right)}}}{1 + \frac{\sin^{2}{\left(t + 90 \right)}}{4 \sin^{4}{\left(\frac{t}{2} + 45 \right)}}} & \text{otherwise} \end{cases} 
/                          1                            for (90 + t - 28*pi) mod 2*pi = 0
|                                                                                        
|/        1          for (90 + t - 28*pi) mod 2*pi = 0                                   
||                                                                                       
||        2/     t\                                                                      
<|-1 + cot |45 + -|                                                                      
|<         \     2/                                                 otherwise            
||-----------------              otherwise                                               
||        2/     t\                                                                      
|| 1 + cot |45 + -|                                                                      
\\         \     2/
{1for (t−28π+90) mod 2π=0{1for (t−28π+90) mod 2π=0cot⁡2(t2+45)−1cot⁡2(t2+45)+1otherwiseotherwise\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} + 45 \right)} - 1}{\cot^{2}{\left(\frac{t}{2} + 45 \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases} 
/          1             for (90 + t - 28*pi) mod 2*pi = 0
|                                                         
|           2/     t\                                     
|        cos |45 + -|                                     
|            \     2/                                     
|-1 + -----------------                                   
|        2/     t   pi\                                   
|     cos |45 + - - --|                                   
<         \     2   2 /                                   
|----------------------              otherwise            
|          2/     t\                                      
|       cos |45 + -|                                      
|           \     2/                                      
|1 + -----------------                                    
|       2/     t   pi\                                    
|    cos |45 + - - --|                                    
\        \     2   2 /
{1for (t−28π+90) mod 2π=0cos⁡2(t2+45)cos⁡2(t2−π2+45)−1cos⁡2(t2+45)cos⁡2(t2−π2+45)+1otherwise\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{t}{2} + 45 \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} + 45 \right)}} - 1}{\frac{\cos^{2}{\left(\frac{t}{2} + 45 \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} + 45 \right)}} + 1} & \text{otherwise} \end{cases} 
/          1             for (90 + t - 28*pi) mod 2*pi = 0
|                                                         
|        2/     t   pi\                                   
|     sec |45 + - - --|                                   
|         \     2   2 /                                   
|-1 + -----------------                                   
|           2/     t\                                     
|        sec |45 + -|                                     
<            \     2/                                     
|----------------------              otherwise            
|       2/     t   pi\                                    
|    sec |45 + - - --|                                    
|        \     2   2 /                                    
|1 + -----------------                                    
|          2/     t\                                      
|       sec |45 + -|                                      
\           \     2/
{1for (t−28π+90) mod 2π=0−1+sec⁡2(t2−π2+45)sec⁡2(t2+45)1+sec⁡2(t2−π2+45)sec⁡2(t2+45)otherwise\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} + 45 \right)}}{\sec^{2}{\left(\frac{t}{2} + 45 \right)}}}{1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} + 45 \right)}}{\sec^{2}{\left(\frac{t}{2} + 45 \right)}}} & \text{otherwise} \end{cases} 
/           1             for (90 + t - 28*pi) mod 2*pi = 0
|                                                          
|           2/     t\                                      
|        csc |45 + -|                                      
|            \     2/                                      
|-1 + ------------------                                   
|        2/      pi   t\                                   
|     csc |-45 + -- - -|                                   
<         \      2    2/                                   
|-----------------------              otherwise            
|           2/     t\                                      
|        csc |45 + -|                                      
|            \     2/                                      
| 1 + ------------------                                   
|        2/      pi   t\                                   
|     csc |-45 + -- - -|                                   
\         \      2    2/
{1for (t−28π+90) mod 2π=0csc⁡2(t2+45)csc⁡2(−t2−45+π2)−1csc⁡2(t2+45)csc⁡2(−t2−45+π2)+1otherwise\begin{cases} 1 & \text{for}\: \left(t - 28 \pi + 90\right) \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{t}{2} + 45 \right)}}{\csc^{2}{\left(- \frac{t}{2} - 45 + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{t}{2} + 45 \right)}}{\csc^{2}{\left(- \frac{t}{2} - 45 + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases} 
Piecewise((1, Mod(90 + t - 28*pi = 2*pi, 0)), ((-1 + csc(45 + t/2)^2/csc(-45 + pi/2 - t/2)^2)/(1 + csc(45 + t/2)^2/csc(-45 + pi/2 - t/2)^2), True))
Степени [src] 
 I*(-90 - t)    I*(90 + t)
e              e          
------------ + -----------
     2              2
ei(−t−90)2+ei(t+90)2\frac{e^{i \left(- t - 90\right)}}{2} + \frac{e^{i \left(t + 90\right)}}{2} 
exp(i*(-90 - t))/2 + exp(i*(90 + t))/2
Численный ответ [src] 
cos(90 + t)
cos(90 + t)
    © Господин Экзамен – Калькулятор