2*sin(15*x)*cos(15*x)еслиx=1/2 (упростите выражение)

Вы ввели

2*sin(15*x)*cos(15*x)

$$2 \sin{\left(15 x \right)} \cos{\left(15 x \right)}$$

Подстановка условия

2*sin(15*x)*cos(15*x) при x = 1/2

подставляем

2*sin(15*x)*cos(15*x)

$$2 \sin{\left(15 x \right)} \cos{\left(15 x \right)}$$

sin(30*x)

$$\sin{\left(30 x \right)}$$

переменные

x = 1/2

$$x = \frac{1}{2}$$

sin(30*(1/2))

$$\sin{\left(30 (1/2) \right)}$$

sin(15)

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

Степени

   / -15*I*x    15*I*x\                       
   |e          e      | /   -15*I*x    15*I*x\
-I*|-------- + -------|*\- e        + e      /
   \   2          2   /                       

$$- i \left(\frac{e^{15 i x}}{2} + \frac{e^{- 15 i x}}{2}\right) \left(e^{15 i x} - e^{- 15 i x}\right)$$

Численный ответ

2.0*cos(15*x)*sin(15*x)

Общее упрощение

sin(30*x)

$$\sin{\left(30 x \right)}$$

Собрать выражение

sin(30*x)

$$\sin{\left (30 x \right )}$$

Тригонометрическая часть

  //      0         for And(im(x) = 0, 15*x mod pi = 0)\ //      1         for And(im(x) = 0, 15*x mod 2*pi = 0)\
  ||                                                   | ||                                                     |
  ||      /15*x\                                       | ||       2/15*x\                                       |
  || 2*tan|----|                                       | ||1 - tan |----|                                       |
2*|<      \ 2  /                                       |*|<        \ 2  /                                       |
  ||--------------               otherwise             | ||--------------                otherwise              |
  ||       2/15*x\                                     | ||       2/15*x\                                       |
  ||1 + tan |----|                                     | ||1 + tan |----|                                       |
  \\        \ 2  /                                     / \\        \ 2  /                                       /

$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{15 x}{2} \right)}}{\tan^{2}{\left(\frac{15 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod 2 \pi = 0 \\\frac{1 - \tan^{2}{\left(\frac{15 x}{2} \right)}}{\tan^{2}{\left(\frac{15 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$

                                                    //      1         for And(im(x) = 0, 15*x mod 2*pi = 0)\
  //    0      for And(im(x) = 0, 15*x mod pi = 0)\ ||                                                     |
2*|<                                              |*|<   /pi       \                                       |
  \\sin(15*x)               otherwise             / ||sin|-- + 15*x|                otherwise              |
                                                    \\   \2        /                                       /

$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod \pi = 0 \\\sin{\left(15 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod 2 \pi = 0 \\\sin{\left(15 x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$

               /       pi\
2*cos(15*x)*cos|15*x - --|
               \       2 /

$$2 \cos{\left(15 x \right)} \cos{\left(15 x - \frac{\pi}{2} \right)}$$

           2            
------------------------
             /       pi\
sec(15*x)*sec|15*x - --|
             \       2 /

$$\frac{2}{\sec{\left(15 x \right)} \sec{\left(15 x - \frac{\pi}{2} \right)}}$$

         2         
-------------------
csc(15*x)*sec(15*x)

$$\frac{2}{\csc{\left(15 x \right)} \sec{\left(15 x \right)}}$$

  //                         0                            for And(im(x) = 0, 15*x mod pi = 0)\ //                           1                             for And(im(x) = 0, 15*x mod 2*pi = 0)\
  ||                                                                                         | ||                                                                                              |
  ||/      0         for And(im(x) = 0, 15*x mod pi = 0)                                     | ||/       1         for And(im(x) = 0, 15*x mod 2*pi = 0)                                       |
  |||                                                                                        | |||                                                                                             |
  |||      /15*x\                                                                            | |||        2/15*x\                                                                              |
2*|<| 2*cot|----|                                                                            |*|<|-1 + cot |----|                                                                              |
  ||<      \ 2  /                                                      otherwise             | ||<         \ 2  /                                                       otherwise              |
  |||--------------               otherwise                                                  | |||---------------                otherwise                                                     |
  |||       2/15*x\                                                                          | |||        2/15*x\                                                                              |
  |||1 + cot |----|                                                                          | ||| 1 + cot |----|                                                                              |
  \\\        \ 2  /                                                                          / \\\         \ 2  /                                                                              /

$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{15 x}{2} \right)}}{\cot^{2}{\left(\frac{15 x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{15 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{15 x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$

    1    
---------
csc(30*x)

$$\frac{1}{\csc{\left(30 x \right)}}$$

  /       2/15*x\\    /15*x\
4*|1 - tan |----||*tan|----|
  \        \ 2  //    \ 2  /
----------------------------
                     2      
     /       2/15*x\\       
     |1 + tan |----||       
     \        \ 2  //       

$$\frac{4 \cdot \left(1 - \tan^{2}{\left(\frac{15 x}{2} \right)}\right) \tan{\left(\frac{15 x}{2} \right)}}{\left(\tan^{2}{\left(\frac{15 x}{2} \right)} + 1\right)^{2}}$$

  //    0      for And(im(x) = 0, 15*x mod pi = 0)\ //    1      for And(im(x) = 0, 15*x mod 2*pi = 0)\
2*|<                                              |*|<                                                |
  \\sin(15*x)               otherwise             / \\cos(15*x)                otherwise              /

$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod \pi = 0 \\\sin{\left(15 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod 2 \pi = 0 \\\cos{\left(15 x \right)} & \text{otherwise} \end{cases}\right)$$

      1       
--------------
   /       pi\
sec|30*x - --|
   \       2 /

$$\frac{1}{\sec{\left(30 x - \frac{\pi}{2} \right)}}$$

               /pi       \
2*sin(15*x)*sin|-- + 15*x|
               \2        /

$$2 \sin{\left(15 x \right)} \sin{\left(15 x + \frac{\pi}{2} \right)}$$

sin(30*x)

$$\sin{\left(30 x \right)}$$

           2            
------------------------
             /pi       \
csc(15*x)*csc|-- - 15*x|
             \2        /

$$\frac{2}{\csc{\left(15 x \right)} \csc{\left(- 15 x + \frac{\pi}{2} \right)}}$$

/    0      for And(im(x) = 0, 30*x mod pi = 0)
<                                              
\sin(30*x)               otherwise             

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 30 x \bmod \pi = 0 \\\sin{\left(30 x \right)} & \text{otherwise} \end{cases}$$

 2*tan(15*x)  
--------------
       2      
1 + tan (15*x)

$$\frac{2 \tan{\left(15 x \right)}}{\tan^{2}{\left(15 x \right)} + 1}$$

                                                    //      1         for And(im(x) = 0, 15*x mod 2*pi = 0)\
  //    0      for And(im(x) = 0, 15*x mod pi = 0)\ ||                                                     |
  ||                                              | ||      1                                              |
2*|<    1                                         |*|<--------------                otherwise              |
  ||---------               otherwise             | ||   /pi       \                                       |
  \\csc(15*x)                                     / ||csc|-- - 15*x|                                       |
                                                    \\   \2        /                                       /

$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod \pi = 0 \\\frac{1}{\csc{\left(15 x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- 15 x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$

/      0         for And(im(x) = 0, 30*x mod pi = 0)
|                                                   
| 2*cot(15*x)                                       
<--------------               otherwise             
|       2                                           
|1 + cot (15*x)                                     
\                                                   

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 30 x \bmod \pi = 0 \\\frac{2 \cot{\left(15 x \right)}}{\cot^{2}{\left(15 x \right)} + 1} & \text{otherwise} \end{cases}$$

  //      0         for And(im(x) = 0, 15*x mod pi = 0)\ //       1         for And(im(x) = 0, 15*x mod 2*pi = 0)\
  ||                                                   | ||                                                      |
  ||      /15*x\                                       | ||        2/15*x\                                       |
  || 2*cot|----|                                       | ||-1 + cot |----|                                       |
2*|<      \ 2  /                                       |*|<         \ 2  /                                       |
  ||--------------               otherwise             | ||---------------                otherwise              |
  ||       2/15*x\                                     | ||        2/15*x\                                       |
  ||1 + cot |----|                                     | || 1 + cot |----|                                       |
  \\        \ 2  /                                     / \\         \ 2  /                                       /

$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{15 x}{2} \right)}}{\cot^{2}{\left(\frac{15 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{15 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{15 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$

   /       pi\
cos|30*x - --|
   \       2 /

$$\cos{\left(30 x - \frac{\pi}{2} \right)}$$

  //                       0                         for And(im(x) = 0, 15*x mod pi = 0)\ //                        1                          for And(im(x) = 0, 15*x mod 2*pi = 0)\
  ||                                                                                    | ||                                                                                        |
2*|

$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod \pi = 0 \\\sin{\left(15 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod 2 \pi = 0 \\\cos{\left(15 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$

  //      0         for And(im(x) = 0, 15*x mod pi = 0)\                                                    
  ||                                                   | //    1      for And(im(x) = 0, 15*x mod 2*pi = 0)\
  ||      1                                            | ||                                                |
2*|<--------------               otherwise             |*|<    1                                           |
  ||   /       pi\                                     | ||---------                otherwise              |
  ||sec|15*x - --|                                     | \\sec(15*x)                                       /
  \\   \       2 /                                     /                                                    

$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod \pi = 0 \\\frac{1}{\sec{\left(15 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(15 x \right)}} & \text{otherwise} \end{cases}\right)$$

  //      0         for And(im(x) = 0, 15*x mod pi = 0)\                                                    
  ||                                                   | //    1      for And(im(x) = 0, 15*x mod 2*pi = 0)\
2*|<   /       pi\                                     |*|<                                                |
  ||cos|15*x - --|               otherwise             | \\cos(15*x)                otherwise              /
  \\   \       2 /                                     /                                                    

$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod \pi = 0 \\\cos{\left(15 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 15 x \bmod 2 \pi = 0 \\\cos{\left(15 x \right)} & \text{otherwise} \end{cases}\right)$$

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- 16986931200*cos  (x)*sin  (x) - 9912320000*cos (x)*sin (x) - 8650752000*cos (x)*sin  (x) - 8650752000*cos  (x)*sin (x) - 7549747200*cos  (x)*sin  (x) - 5308416000*cos (x)*sin  (x) - 5308416000*cos  (x)*sin (x) - 3019898880*cos  (x)*sin  (x) - 3019898880*cos  (x)*sin  (x) - 1658880000*cos (x)*sin (x) - 943718400*cos (x)*sin  (x) - 943718400*cos  (x)*sin (x) - 851558400*cos (x)*sin (x) - 851558400*cos (x)*sin (x) - 743178240*cos (x)*sin  (x) - 743178240*cos  (x)*sin (x) - 536870912*cos  (x)*sin  (x) - 103219200*cos (x)*sin  (x) - 103219200*cos  (x)*sin (x) - 73156608*cos (x)*sin (x) - 32256000*cos (x)*sin (x) - 32256000*cos (x)*sin (x) - 18350080*cos (x)*sin  (x) - 18350080*cos  (x)*sin (x) - 2112000*cos (x)*sin(x) - 2112000*sin (x)*cos(x) - 1843200*cos  (x)*sin(x) - 1843200*sin  (x)*cos(x) - 627200*cos (x)*sin (x) - 181440*cos (x)*sin(x) - 181440*sin (x)*cos(x) - 450*cos(x)*sin(x) + 16800*cos (x)*sin(x) + 16800*sin (x)*cos(x) + 491520*cos  (x)*sin(x) + 491520*sin  (x)*cos(x) + 864000*cos (x)*sin(x) + 864000*sin (x)*cos(x) + 2764800*cos  (x)*sin(x) + 2764800*sin  (x)*cos(x) + 6773760*cos (x)*sin (x) + 6773760*cos (x)*sin (x) + 68812800*cos (x)*sin  (x) + 68812800*cos  (x)*sin (x) + 78848000*cos (x)*sin (x) + 78848000*cos (x)*sin (x) + 198180864*cos (x)*sin  (x) + 198180864*cos  (x)*sin (x) + 348364800*cos (x)*sin (x) + 348364800*cos (x)*sin (x) + 1114767360*cos (x)*sin  (x) + 1114767360*cos  (x)*sin (x) + 2013265920*cos  (x)*sin  (x) + 2013265920*cos  (x)*sin  (x) + 2306867200*cos (x)*sin  (x) + 2306867200*cos  (x)*sin (x) + 3538944000*cos (x)*sin  (x) + 3538944000*cos  (x)*sin (x) + 4055040000*cos (x)*sin (x) + 4055040000*cos (x)*sin (x) + 11324620800*cos  (x)*sin  (x) + 11324620800*cos  (x)*sin  (x) + 12976128000*cos (x)*sin  (x) + 12976128000*cos  (x)*sin (x)

$$- 536870912 \sin^{15}{\left(x \right)} \cos^{15}{\left(x \right)} + 2013265920 \sin^{15}{\left(x \right)} \cos^{13}{\left(x \right)} - 3019898880 \sin^{15}{\left(x \right)} \cos^{11}{\left(x \right)} + 2306867200 \sin^{15}{\left(x \right)} \cos^{9}{\left(x \right)} - 943718400 \sin^{15}{\left(x \right)} \cos^{7}{\left(x \right)} + 198180864 \sin^{15}{\left(x \right)} \cos^{5}{\left(x \right)} - 18350080 \sin^{15}{\left(x \right)} \cos^{3}{\left(x \right)} + 491520 \sin^{15}{\left(x \right)} \cos{\left(x \right)} + 2013265920 \sin^{13}{\left(x \right)} \cos^{15}{\left(x \right)} - 7549747200 \sin^{13}{\left(x \right)} \cos^{13}{\left(x \right)} + 11324620800 \sin^{13}{\left(x \right)} \cos^{11}{\left(x \right)} - 8650752000 \sin^{13}{\left(x \right)} \cos^{9}{\left(x \right)} + 3538944000 \sin^{13}{\left(x \right)} \cos^{7}{\left(x \right)} - 743178240 \sin^{13}{\left(x \right)} \cos^{5}{\left(x \right)} + 68812800 \sin^{13}{\left(x \right)} \cos^{3}{\left(x \right)} - 1843200 \sin^{13}{\left(x \right)} \cos{\left(x \right)} - 3019898880 \sin^{11}{\left(x \right)} \cos^{15}{\left(x \right)} + 11324620800 \sin^{11}{\left(x \right)} \cos^{13}{\left(x \right)} - 16986931200 \sin^{11}{\left(x \right)} \cos^{11}{\left(x \right)} + 12976128000 \sin^{11}{\left(x \right)} \cos^{9}{\left(x \right)} - 5308416000 \sin^{11}{\left(x \right)} \cos^{7}{\left(x \right)} + 1114767360 \sin^{11}{\left(x \right)} \cos^{5}{\left(x \right)} - 103219200 \sin^{11}{\left(x \right)} \cos^{3}{\left(x \right)} + 2764800 \sin^{11}{\left(x \right)} \cos{\left(x \right)} + 2306867200 \sin^{9}{\left(x \right)} \cos^{15}{\left(x \right)} - 8650752000 \sin^{9}{\left(x \right)} \cos^{13}{\left(x \right)} + 12976128000 \sin^{9}{\left(x \right)} \cos^{11}{\left(x \right)} - 9912320000 \sin^{9}{\left(x \right)} \cos^{9}{\left(x \right)} + 4055040000 \sin^{9}{\left(x \right)} \cos^{7}{\left(x \right)} - 851558400 \sin^{9}{\left(x \right)} \cos^{5}{\left(x \right)} + 78848000 \sin^{9}{\left(x \right)} \cos^{3}{\left(x \right)} - 2112000 \sin^{9}{\left(x \right)} \cos{\left(x \right)} - 943718400 \sin^{7}{\left(x \right)} \cos^{15}{\left(x \right)} + 3538944000 \sin^{7}{\left(x \right)} \cos^{13}{\left(x \right)} - 5308416000 \sin^{7}{\left(x \right)} \cos^{11}{\left(x \right)} + 4055040000 \sin^{7}{\left(x \right)} \cos^{9}{\left(x \right)} - 1658880000 \sin^{7}{\left(x \right)} \cos^{7}{\left(x \right)} + 348364800 \sin^{7}{\left(x \right)} \cos^{5}{\left(x \right)} - 32256000 \sin^{7}{\left(x \right)} \cos^{3}{\left(x \right)} + 864000 \sin^{7}{\left(x \right)} \cos{\left(x \right)} + 198180864 \sin^{5}{\left(x \right)} \cos^{15}{\left(x \right)} - 743178240 \sin^{5}{\left(x \right)} \cos^{13}{\left(x \right)} + 1114767360 \sin^{5}{\left(x \right)} \cos^{11}{\left(x \right)} - 851558400 \sin^{5}{\left(x \right)} \cos^{9}{\left(x \right)} + 348364800 \sin^{5}{\left(x \right)} \cos^{7}{\left(x \right)} - 73156608 \sin^{5}{\left(x \right)} \cos^{5}{\left(x \right)} + 6773760 \sin^{5}{\left(x \right)} \cos^{3}{\left(x \right)} - 181440 \sin^{5}{\left(x \right)} \cos{\left(x \right)} - 18350080 \sin^{3}{\left(x \right)} \cos^{15}{\left(x \right)} + 68812800 \sin^{3}{\left(x \right)} \cos^{13}{\left(x \right)} - 103219200 \sin^{3}{\left(x \right)} \cos^{11}{\left(x \right)} + 78848000 \sin^{3}{\left(x \right)} \cos^{9}{\left(x \right)} - 32256000 \sin^{3}{\left(x \right)} \cos^{7}{\left(x \right)} + 6773760 \sin^{3}{\left(x \right)} \cos^{5}{\left(x \right)} - 627200 \sin^{3}{\left(x \right)} \cos^{3}{\left(x \right)} + 16800 \sin^{3}{\left(x \right)} \cos{\left(x \right)} + 491520 \sin{\left(x \right)} \cos^{15}{\left(x \right)} - 1843200 \sin{\left(x \right)} \cos^{13}{\left(x \right)} + 2764800 \sin{\left(x \right)} \cos^{11}{\left(x \right)} - 2112000 \sin{\left(x \right)} \cos^{9}{\left(x \right)} + 864000 \sin{\left(x \right)} \cos^{7}{\left(x \right)} - 181440 \sin{\left(x \right)} \cos^{5}{\left(x \right)} + 16800 \sin{\left(x \right)} \cos^{3}{\left(x \right)} - 450 \sin{\left(x \right)} \cos{\left(x \right)}$$