sin(5*x)*sin(3*x)еслиx=-3 (упростите выражение)

Вы ввели

sin(5*x)*sin(3*x)

$$\sin{\left(3 x \right)} \sin{\left(5 x \right)}$$

Подстановка условия

sin(5*x)*sin(3*x) при x = -3

подставляем

sin(5*x)*sin(3*x)

$$\sin{\left(3 x \right)} \sin{\left(5 x \right)}$$

sin(3*x)*sin(5*x)

$$\sin{\left(3 x \right)} \sin{\left(5 x \right)}$$

переменные

x = -3

$$x = -3$$

sin(3*(-3))*sin(5*(-3))

$$\sin{\left(3 (-3) \right)} \sin{\left(5 (-3) \right)}$$

sin(9)*sin(15)

$$\sin{\left(9 \right)} \sin{\left(15 \right)}$$

Степени

 /   -5*I*x    5*I*x\ /   -3*I*x    3*I*x\ 
-\- e       + e     /*\- e       + e     / 
-------------------------------------------
                     4                     

$$- \frac{\left(e^{3 i x} - e^{- 3 i x}\right) \left(e^{5 i x} - e^{- 5 i x}\right)}{4}$$

Численный ответ

sin(3*x)*sin(5*x)

Собрать выражение

cos(2*x)   cos(8*x)
-------- - --------
   2          2    

$$\frac{1}{2} \cos{\left (2 x \right )} - \frac{1}{2} \cos{\left (8 x \right )}$$

Тригонометрическая часть

//      0        for And(im(x) = 0, 3*x mod pi = 0)\ //      0        for And(im(x) = 0, 5*x mod pi = 0)\
||                                                 | ||                                                 |
||      1                                          | ||      1                                          |
|<-------------              otherwise             |*|<-------------              otherwise             |
||   /      pi\                                    | ||   /      pi\                                    |
||sec|3*x - --|                                    | ||sec|5*x - --|                                    |
\\   \      2 /                                    / \\   \      2 /                                    /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\frac{1}{\sec{\left(3 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 5 x \bmod \pi = 0 \\\frac{1}{\sec{\left(5 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$

    1            1     
---------- - ----------
2*sec(2*x)   2*sec(8*x)

$$- \frac{1}{2 \sec{\left(8 x \right)}} + \frac{1}{2 \sec{\left(2 x \right)}}$$

   /pi      \      /pi      \
sin|-- + 2*x|   sin|-- + 8*x|
   \2       /      \2       /
------------- - -------------
      2               2      

$$\frac{\sin{\left(2 x + \frac{\pi}{2} \right)}}{2} - \frac{\sin{\left(8 x + \frac{\pi}{2} \right)}}{2}$$

           /3*x\    /5*x\      
      4*tan|---|*tan|---|      
           \ 2 /    \ 2 /      
-------------------------------
/       2/3*x\\ /       2/5*x\\
|1 + tan |---||*|1 + tan |---||
\        \ 2 // \        \ 2 //

$$\frac{4 \tan{\left(\frac{3 x}{2} \right)} \tan{\left(\frac{5 x}{2} \right)}}{\left(\tan^{2}{\left(\frac{3 x}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{5 x}{2} \right)} + 1\right)}$$

cos(2*x)   cos(8*x)
-------- - --------
   2          2    

$$\frac{\cos{\left(2 x \right)}}{2} - \frac{\cos{\left(8 x \right)}}{2}$$

                                              /            1               for And(im(x) = 0, 4*x mod pi = 0)
                                              |                                                              
/   1      for And(im(x) = 0, x mod pi = 0)   <   2      /        2     \                                    
<                                             |sin (4*x)*\-1 + cot (4*x)/              otherwise             
\cos(2*x)             otherwise               \                                                              
------------------------------------------- - ---------------------------------------------------------------
                     2                                                       2                               

$$\left(\frac{\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}}{2}\right) - \left(\frac{\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 4 x \bmod \pi = 0 \\\left(\cot^{2}{\left(4 x \right)} - 1\right) \sin^{2}{\left(4 x \right)} & \text{otherwise} \end{cases}}{2}\right)$$

//      0        for And(im(x) = 0, 3*x mod pi = 0)\ //      0        for And(im(x) = 0, 5*x mod pi = 0)\
||                                                 | ||                                                 |
|<   /      pi\                                    |*|<   /      pi\                                    |
||cos|3*x - --|              otherwise             | ||cos|5*x - --|              otherwise             |
\\   \      2 /                                    / \\   \      2 /                                    /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\cos{\left(3 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 5 x \bmod \pi = 0 \\\cos{\left(5 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$

       1                 1       
--------------- - ---------------
     /pi      \        /pi      \
2*csc|-- - 2*x|   2*csc|-- - 8*x|
     \2       /        \2       /

$$\frac{1}{2 \csc{\left(- 2 x + \frac{\pi}{2} \right)}} - \frac{1}{2 \csc{\left(- 8 x + \frac{\pi}{2} \right)}}$$

/     1        for And(im(x) = 0, x mod pi = 0)   /      1         for And(im(x) = 0, 4*x mod pi = 0)
|                                                 |                                                  
|        2                                        |        2                                         
<-1 + cot (x)                                     <-1 + cot (4*x)                                    
|------------             otherwise               |--------------              otherwise             
|       2                                         |       2                                          
\1 + cot (x)                                      \1 + cot (4*x)                                     
----------------------------------------------- - ---------------------------------------------------
                       2                                                   2                         

$$\left(\frac{\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) - \left(\frac{\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 4 x \bmod \pi = 0 \\\frac{\cot^{2}{\left(4 x \right)} - 1}{\cot^{2}{\left(4 x \right)} + 1} & \text{otherwise} \end{cases}}{2}\right)$$

//                      0                        for And(im(x) = 0, 3*x mod pi = 0)\ //                      0                        for And(im(x) = 0, 5*x mod pi = 0)\
||                                                                                 | ||                                                                                 |
|

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\sin{\left(3 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 5 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 5 x \bmod \pi = 0 \\\sin{\left(5 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$

//      0        for And(im(x) = 0, 3*x mod pi = 0)\ //      0        for And(im(x) = 0, 5*x mod pi = 0)\
||                                                 | ||                                                 |
||       /3*x\                                     | ||       /5*x\                                     |
||  2*tan|---|                                     | ||  2*tan|---|                                     |
|<       \ 2 /                                     |*|<       \ 2 /                                     |
||-------------              otherwise             | ||-------------              otherwise             |
||       2/3*x\                                    | ||       2/5*x\                                    |
||1 + tan |---|                                    | ||1 + tan |---|                                    |
\\        \ 2 /                                    / \\        \ 2 /                                    /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{3 x}{2} \right)}}{\tan^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 5 x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{5 x}{2} \right)}}{\tan^{2}{\left(\frac{5 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$

//                        0                           for And(im(x) = 0, 3*x mod pi = 0)\ //                        0                           for And(im(x) = 0, 5*x mod pi = 0)\
||                                                                                      | ||                                                                                      |
||/      0        for And(im(x) = 0, 3*x mod pi = 0)                                    | ||/      0        for And(im(x) = 0, 5*x mod pi = 0)                                    |
|||                                                                                     | |||                                                                                     |
|||       /3*x\                                                                         | |||       /5*x\                                                                         |
|<|  2*cot|---|                                                                         |*|<|  2*cot|---|                                                                         |
||<       \ 2 /                                                   otherwise             | ||<       \ 2 /                                                   otherwise             |
|||-------------              otherwise                                                 | |||-------------              otherwise                                                 |
|||       2/3*x\                                                                        | |||       2/5*x\                                                                        |
|||1 + cot |---|                                                                        | |||1 + cot |---|                                                                        |
\\\        \ 2 /                                                                        / \\\        \ 2 /                                                                        /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 x}{2} \right)}}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 5 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 5 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{5 x}{2} \right)}}{\cot^{2}{\left(\frac{5 x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$

//      0        for And(im(x) = 0, 3*x mod pi = 0)\ //      0        for And(im(x) = 0, 5*x mod pi = 0)\
||                                                 | ||                                                 |
||       /3*x\                                     | ||       /5*x\                                     |
||  2*cot|---|                                     | ||  2*cot|---|                                     |
|<       \ 2 /                                     |*|<       \ 2 /                                     |
||-------------              otherwise             | ||-------------              otherwise             |
||       2/3*x\                                    | ||       2/5*x\                                    |
||1 + cot |---|                                    | ||1 + cot |---|                                    |
\\        \ 2 /                                    / \\        \ 2 /                                    /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 x}{2} \right)}}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 5 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{5 x}{2} \right)}}{\cot^{2}{\left(\frac{5 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$

             1             
---------------------------
   /      pi\    /      pi\
sec|3*x - --|*sec|5*x - --|
   \      2 /    \      2 /

$$\frac{1}{\sec{\left(3 x - \frac{\pi}{2} \right)} \sec{\left(5 x - \frac{\pi}{2} \right)}}$$

        1        
-----------------
csc(3*x)*csc(5*x)

$$\frac{1}{\csc{\left(3 x \right)} \csc{\left(5 x \right)}}$$

              /pi      \
           sin|-- + 8*x|
cos(2*x)      \2       /
-------- - -------------
   2             2      

$$- \frac{\sin{\left(8 x + \frac{\pi}{2} \right)}}{2} + \frac{\cos{\left(2 x \right)}}{2}$$

//   0      for And(im(x) = 0, 3*x mod pi = 0)\ //   0      for And(im(x) = 0, 5*x mod pi = 0)\
||                                            | ||                                            |
|<   1                                        |*|<   1                                        |
||--------              otherwise             | ||--------              otherwise             |
\\csc(3*x)                                    / \\csc(5*x)                                    /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\frac{1}{\csc{\left(3 x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 5 x \bmod \pi = 0 \\\frac{1}{\csc{\left(5 x \right)}} & \text{otherwise} \end{cases}\right)$$

cos(2*x)       1     
-------- - ----------
   2       2*sec(8*x)

$$\frac{\cos{\left(2 x \right)}}{2} - \frac{1}{2 \sec{\left(8 x \right)}}$$

              2      /       2     \
cos(2*x)   cos (4*x)*\1 - tan (4*x)/
-------- - -------------------------
   2                   2            

$$- \frac{\left(1 - \tan^{2}{\left(4 x \right)}\right) \cos^{2}{\left(4 x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{2}$$

         2                 2       
  1 - tan (x)       1 - tan (4*x)  
--------------- - -----------------
  /       2   \     /       2     \
2*\1 + tan (x)/   2*\1 + tan (4*x)/

$$\frac{1 - \tan^{2}{\left(x \right)}}{2 \left(\tan^{2}{\left(x \right)} + 1\right)} - \frac{1 - \tan^{2}{\left(4 x \right)}}{2 \left(\tan^{2}{\left(4 x \right)} + 1\right)}$$

   /      pi\    /      pi\
cos|3*x - --|*cos|5*x - --|
   \      2 /    \      2 /

$$\cos{\left(3 x - \frac{\pi}{2} \right)} \cos{\left(5 x - \frac{\pi}{2} \right)}$$

//   0      for And(im(x) = 0, 3*x mod pi = 0)\ //   0      for And(im(x) = 0, 5*x mod pi = 0)\
|<                                            |*|<                                            |
\\sin(3*x)              otherwise             / \\sin(5*x)              otherwise             /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\sin{\left(3 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 5 x \bmod \pi = 0 \\\sin{\left(5 x \right)} & \text{otherwise} \end{cases}\right)$$

cos(2*x)          1       
-------- - ---------------
   2            /pi      \
           2*csc|-- - 8*x|
                \2       /

$$\frac{\cos{\left(2 x \right)}}{2} - \frac{1}{2 \csc{\left(- 8 x + \frac{\pi}{2} \right)}}$$

Раскрыть выражение

        4            8            2             6   
- 80*sin (x) - 64*sin (x) + 15*sin (x) + 128*sin (x)

$$- 64 \sin^{8}{\left(x \right)} + 128 \sin^{6}{\left(x \right)} - 80 \sin^{4}{\left(x \right)} + 15 \sin^{2}{\left(x \right)}$$