cos(5*n)еслиn=-2 (упростите выражение)

Вы ввели

cos(5*n)

$$\cos{\left(5 n \right)}$$

Подстановка условия

cos(5*n) при n = -2

подставляем

cos(5*n)

$$\cos{\left(5 n \right)}$$

cos(5*n)

$$\cos{\left(5 n \right)}$$

переменные

n = -2

$$n = -2$$

cos(5*(-2))

$$\cos{\left(5 (-2) \right)}$$

cos(10)

$$\cos{\left(10 \right)}$$

Степени

 -5*I*n    5*I*n
e         e     
------- + ------
   2        2   

$$\frac{e^{5 i n}}{2} + \frac{e^{- 5 i n}}{2}$$

Численный ответ

cos(5*n)

Тригонометрическая часть

/      1        for And(im(n) = 0, 5*n mod 2*pi = 0)
|                                                   
|      1                                            
<-------------               otherwise              
|   /pi      \                                      
|csc|-- - 5*n|                                      
\   \2       /                                      

$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n\right)} = 0 \wedge 5 n \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- 5 n + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}$$

       2/5*n\
1 - tan |---|
        \ 2 /
-------------
       2/5*n\
1 + tan |---|
        \ 2 /

$$\frac{1 - \tan^{2}{\left(\frac{5 n}{2} \right)}}{\tan^{2}{\left(\frac{5 n}{2} \right)} + 1}$$

/      1        for And(im(n) = 0, 5*n mod 2*pi = 0)
|                                                   
|       2/5*n\                                      
|1 - tan |---|                                      
<        \ 2 /                                      
|-------------               otherwise              
|       2/5*n\                                      
|1 + tan |---|                                      
\        \ 2 /                                      

$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n\right)} = 0 \wedge 5 n \bmod 2 \pi = 0 \\\frac{1 - \tan^{2}{\left(\frac{5 n}{2} \right)}}{\tan^{2}{\left(\frac{5 n}{2} \right)} + 1} & \text{otherwise} \end{cases}$$

/   1      for And(im(n) = 0, 5*n mod 2*pi = 0)
<                                              
\cos(5*n)               otherwise              

$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n\right)} = 0 \wedge 5 n \bmod 2 \pi = 0 \\\cos{\left(5 n \right)} & \text{otherwise} \end{cases}$$

/                          1                            for And(im(n) = 0, 5*n mod 2*pi = 0)
|                                                                                           
|/      1         for And(im(n) = 0, 5*n mod 2*pi = 0)                                      
||                                                                                          
||        2/5*n\                                                                            
<|-1 + cot |---|                                                                            
|<         \ 2 /                                                     otherwise              
||--------------               otherwise                                                    
||       2/5*n\                                                                             
||1 + cot |---|                                                                             
\\        \ 2 /                                                                             

$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n\right)} = 0 \wedge 5 n \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n\right)} = 0 \wedge 5 n \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{5 n}{2} \right)} - 1}{\cot^{2}{\left(\frac{5 n}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}$$

/   1      for And(im(n) = 0, 5*n mod 2*pi = 0)
|                                              
<   1                                          
|--------               otherwise              
\sec(5*n)                                      

$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n\right)} = 0 \wedge 5 n \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(5 n \right)}} & \text{otherwise} \end{cases}$$

      1      
-------------
   /pi      \
csc|-- - 5*n|
   \2       /

$$\frac{1}{\csc{\left(- 5 n + \frac{\pi}{2} \right)}}$$

/                       1                         for And(im(n) = 0, 5*n mod 2*pi = 0)
|                                                                                     

$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n\right)} = 0 \wedge 5 n \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n\right)} = 0 \wedge 5 n \bmod 2 \pi = 0 \\\cos{\left(5 n \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}$$

   1    
--------
sec(5*n)

$$\frac{1}{\sec{\left(5 n \right)}}$$

/      1        for And(im(n) = 0, 5*n mod 2*pi = 0)
|                                                   
<   /pi      \                                      
|sin|-- + 5*n|               otherwise              
\   \2       /                                      

$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n\right)} = 0 \wedge 5 n \bmod 2 \pi = 0 \\\sin{\left(5 n + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}$$

/      1         for And(im(n) = 0, 5*n mod 2*pi = 0)
|                                                    
|        2/5*n\                                      
|-1 + cot |---|                                      
<         \ 2 /                                      
|--------------               otherwise              
|       2/5*n\                                       
|1 + cot |---|                                       
\        \ 2 /                                       

$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n\right)} = 0 \wedge 5 n \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{5 n}{2} \right)} - 1}{\cot^{2}{\left(\frac{5 n}{2} \right)} + 1} & \text{otherwise} \end{cases}$$

   /pi      \
sin|-- + 5*n|
   \2       /

$$\sin{\left(5 n + \frac{\pi}{2} \right)}$$

Раскрыть выражение

        3                       5   
- 20*cos (n) + 5*cos(n) + 16*cos (n)

$$16 \cos^{5}{\left(n \right)} - 20 \cos^{3}{\left(n \right)} + 5 \cos{\left(n \right)}$$