cos(pi+3*t)еслиt=1 (упростите выражение)

Вы ввели

cos(pi + 3*t)

$$\cos{\left(3 t + \pi \right)}$$

Подстановка условия

cos(pi + 3*t) при t = 1

подставляем

cos(pi + 3*t)

$$\cos{\left(3 t + \pi \right)}$$

-cos(3*t)

$$- \cos{\left(3 t \right)}$$

переменные

t = 1

$$t = 1$$

-cos(3*(1))

$$- \cos{\left(3 (1) \right)}$$

-cos(3*1)

$$- \cos{\left(3 \cdot 1 \right)}$$

-cos(3)

$$- \cos{\left(3 \right)}$$

Степени

 I*(pi + 3*t)    I*(-pi - 3*t)
e               e             
------------- + --------------
      2               2       

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

-cos(3*t)

$$- \cos{\left(3 t \right)}$$

Численный ответ

cos(pi + 3*t)

Рациональный знаменатель

-cos(3*t)

$$- \cos{\left(3 t \right)}$$

Объединение рациональных выражений

-cos(3*t)

$$- \cos{\left(3 t \right)}$$

Общее упрощение

-cos(3*t)

$$- \cos{\left(3 t \right)}$$

Тригонометрическая часть

-cos(3*t)

$$- \cos{\left(3 t \right)}$$

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

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

    /pi      \
-sin|-- + 3*t|
    \2       /

$$- \sin{\left(3 t + \frac{\pi}{2} \right)}$$

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

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

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

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

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

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

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

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

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

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

     -1      
-------------
   /pi      \
csc|-- - 3*t|
   \2       /

$$- \frac{1}{\csc{\left(- 3 t + \frac{\pi}{2} \right)}}$$

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

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

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

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

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

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

   /      3*pi\
sin|3*t + ----|
   \       2  /

$$\sin{\left(3 t + \frac{3 \pi}{2} \right)}$$

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

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

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

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

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

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

  -1    
--------
sec(3*t)

$$- \frac{1}{\sec{\left(3 t \right)}}$$

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

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

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

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

Комбинаторика

-cos(3*t)

$$- \cos{\left(3 t \right)}$$

Раскрыть выражение

       3              
- 4*cos (t) + 3*cos(t)

$$- 4 \cos^{3}{\left(t \right)} + 3 \cos{\left(t \right)}$$