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8-5*cos(2*x)-5*sin(2*x)еслиx=3 (упростите выражение)

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Решение

Вы ввели [src]
8 - 5*cos(2*x) - 5*sin(2*x)
−5sin⁡(2x)−5cos⁡(2x)+8- 5 \sin{\left(2 x \right)} - 5 \cos{\left(2 x \right)} + 8
Подстановка условия [src]
8 - 5*cos(2*x) - 5*sin(2*x) при x = 3
подставляем
8 - 5*cos(2*x) - 5*sin(2*x)
−5sin⁡(2x)−5cos⁡(2x)+8- 5 \sin{\left(2 x \right)} - 5 \cos{\left(2 x \right)} + 8
        ___    /      pi\
8 - 5*\/ 2 *sin|2*x + --|
               \      4 /
−52sin⁡(2x+π4)+8- 5 \sqrt{2} \sin{\left(2 x + \frac{\pi}{4} \right)} + 8
переменные
x = 3
x=3x = 3
        ___    /        pi\
8 - 5*\/ 2 *sin|2*(3) + --|
               \        4 /
−52sin⁡(2(3)+π4)+8- 5 \sqrt{2} \sin{\left(2 (3) + \frac{\pi}{4} \right)} + 8
        ___    /      pi\
8 - 5*\/ 2 *sin|2*3 + --|
               \      4 /
−52sin⁡(π4+2⋅3)+8- 5 \sqrt{2} \sin{\left(\frac{\pi}{4} + 2 \cdot 3 \right)} + 8
        ___    /    pi\
8 - 5*\/ 2 *sin|6 + --|
               \    4 /
−52sin⁡(π4+6)+8- 5 \sqrt{2} \sin{\left(\frac{\pi}{4} + 6 \right)} + 8
Степени [src]
       -2*I*x      2*I*x       /   -2*I*x    2*I*x\
    5*e         5*e        5*I*\- e       + e     /
8 - --------- - -------- + ------------------------
        2          2                  2
5i(e2ix−e−2ix)2−5e2ix2+8−5e−2ix2\frac{5 i \left(e^{2 i x} - e^{- 2 i x}\right)}{2} - \frac{5 e^{2 i x}}{2} + 8 - \frac{5 e^{- 2 i x}}{2}
Численный ответ [src]
8.0 - 5.0*cos(2*x) - 5.0*sin(2*x)
Рациональный знаменатель [src]
8 - 5*cos(2*x) - 5*sin(2*x)
−5sin⁡(2x)−5cos⁡(2x)+8- 5 \sin{\left (2 x \right )} - 5 \cos{\left (2 x \right )} + 8
Объединение рациональных выражений [src]
8 - 5*cos(2*x) - 5*sin(2*x)
−5sin⁡(2x)−5cos⁡(2x)+8- 5 \sin{\left (2 x \right )} - 5 \cos{\left (2 x \right )} + 8
Общее упрощение [src]
        ___    /      pi\
8 - 5*\/ 2 *sin|2*x + --|
               \      4 /
−52sin⁡(2x+π4)+8- 5 \sqrt{2} \sin{\left(2 x + \frac{\pi}{4} \right)} + 8
Собрать выражение [src]
8 - 5*cos(2*x) - 5*sin(2*x)
−5sin⁡(2x)−5cos⁡(2x)+8- 5 \sin{\left (2 x \right )} - 5 \cos{\left (2 x \right )} + 8
Общий знаменатель [src]
8 - 5*cos(2*x) - 5*sin(2*x)
−5sin⁡(2x)−5cos⁡(2x)+8- 5 \sin{\left (2 x \right )} - 5 \cos{\left (2 x \right )} + 8
Тригонометрическая часть [src]
                      /      pi\
8 - 5*cos(2*x) - 5*cos|2*x - --|
                      \      2 /
−5cos⁡(2x)−5cos⁡(2x−π2)+8- 5 \cos{\left(2 x \right)} - 5 \cos{\left(2 x - \frac{\pi}{2} \right)} + 8
      //                      0                        for And(im(x) = 0, 2*x mod pi = 0)\     //                     1                       for And(im(x) = 0, x mod pi = 0)\
      ||                                                                                 |     ||                                                                             |
8 - 5*|
(−5({0for im⁡(x)=0∧2x mod π=0{0for im⁡(x)=0∧2x mod π=0sin⁡(2x)otherwiseotherwise))−(5({1for im⁡(x)=0∧x mod π=0{1for im⁡(x)=0∧x mod π=0cos⁡(2x)otherwiseotherwise))+8\left(- 5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\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} & \text{otherwise} \end{cases}\right)\right) + 8
                      /pi      \
8 - 5*sin(2*x) - 5*sin|-- + 2*x|
                      \2       /
−5sin⁡(2x)−5sin⁡(2x+π2)+8- 5 \sin{\left(2 x \right)} - 5 \sin{\left(2 x + \frac{\pi}{2} \right)} + 8
       5             5      
8 - -------- - -------------
    csc(2*x)      /pi      \
               csc|-- - 2*x|
                  \2       /
8−5csc⁡(−2x+π2)−5csc⁡(2x)8 - \frac{5}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} - \frac{5}{\csc{\left(2 x \right)}}
      //   0      for And(im(x) = 0, 2*x mod pi = 0)\     //   1      for And(im(x) = 0, x mod pi = 0)\
8 - 5*|<                                            | - 5*|<                                          |
      \\sin(2*x)              otherwise             /     \\cos(2*x)             otherwise            /
(−5({0for im⁡(x)=0∧2x mod π=0sin⁡(2x)otherwise))−(5({1for im⁡(x)=0∧x mod π=0cos⁡(2x)otherwise))+8\left(- 5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\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}\right)\right) + 8
           ___   
       5*\/ 2    
8 - -------------
       /      pi\
    csc|2*x + --|
       \      4 /
8−52csc⁡(2x+π4)8 - \frac{5 \sqrt{2}}{\csc{\left(2 x + \frac{\pi}{4} \right)}}
        ___    /      pi\
8 - 5*\/ 2 *sin|2*x + --|
               \      4 /
−52sin⁡(2x+π4)+8- 5 \sqrt{2} \sin{\left(2 x + \frac{\pi}{4} \right)} + 8
       5             5      
8 - -------- - -------------
    sec(2*x)      /      pi\
               sec|2*x - --|
                  \      2 /
8−5sec⁡(2x−π2)−5sec⁡(2x)8 - \frac{5}{\sec{\left(2 x - \frac{\pi}{2} \right)}} - \frac{5}{\sec{\left(2 x \right)}}
           ___   
       5*\/ 2    
8 - -------------
       /      pi\
    sec|2*x - --|
       \      4 /
8−52sec⁡(2x−π4)8 - \frac{5 \sqrt{2}}{\sec{\left(2 x - \frac{\pi}{4} \right)}}
         ___    /    pi\
    10*\/ 2 *tan|x + --|
                \    8 /
8 - --------------------
             2/    pi\  
      1 + tan |x + --|  
              \    8 /
8−102tan⁡(x+π8)tan⁡2(x+π8)+18 - \frac{10 \sqrt{2} \tan{\left(x + \frac{\pi}{8} \right)}}{\tan^{2}{\left(x + \frac{\pi}{8} \right)} + 1}
                    /       2   \
     10*tan(x)    5*\1 - tan (x)/
8 - ----------- - ---------------
           2               2     
    1 + tan (x)     1 + tan (x)
−5⋅(1−tan⁡2(x))tan⁡2(x)+1+8−10tan⁡(x)tan⁡2(x)+1- \frac{5 \cdot \left(1 - \tan^{2}{\left(x \right)}\right)}{\tan^{2}{\left(x \right)} + 1} + 8 - \frac{10 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}
       5          5    
8 - -------- - --------
    csc(2*x)   sec(2*x)
8−5sec⁡(2x)−5csc⁡(2x)8 - \frac{5}{\sec{\left(2 x \right)}} - \frac{5}{\csc{\left(2 x \right)}}
      //     0       for And(im(x) = 0, 2*x mod pi = 0)\     //     1        for And(im(x) = 0, x mod pi = 0)\
      ||                                               |     ||                                              |
      ||  2*cot(x)                                     |     ||        2                                     |
8 - 5*|<-----------              otherwise             | - 5*|<-1 + cot (x)                                  |
      ||       2                                       |     ||------------             otherwise            |
      ||1 + cot (x)                                    |     ||       2                                      |
      \\                                               /     \\1 + cot (x)                                   /
(−5({0for im⁡(x)=0∧2x mod π=02cot⁡(x)cot⁡2(x)+1otherwise))−(5({1for im⁡(x)=0∧x mod π=0cot⁡2(x)−1cot⁡2(x)+1otherwise))+8\left(- 5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\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}\right)\right) + 8
                                                          //      1        for And(im(x) = 0, x mod pi = 0)\
      //   0      for And(im(x) = 0, 2*x mod pi = 0)\     ||                                               |
8 - 5*|<                                            | - 5*|<   /pi      \                                  |
      \\sin(2*x)              otherwise             /     ||sin|-- + 2*x|             otherwise            |
                                                          \\   \2       /                                  /
(−5({0for im⁡(x)=0∧2x mod π=0sin⁡(2x)otherwise))−(5({1for im⁡(x)=0∧x mod π=0sin⁡(2x+π2)otherwise))+8\left(- 5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin{\left(2 x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + 8
      //     0       for And(im(x) = 0, 2*x mod pi = 0)\     //     1       for And(im(x) = 0, x mod pi = 0)\
      ||                                               |     ||                                             |
      ||  2*tan(x)                                     |     ||       2                                     |
8 - 5*|<-----------              otherwise             | - 5*|<1 - tan (x)                                  |
      ||       2                                       |     ||-----------             otherwise            |
      ||1 + tan (x)                                    |     ||       2                                     |
      \\                                               /     \\1 + tan (x)                                  /
(−5({0for im⁡(x)=0∧2x mod π=02tan⁡(x)tan⁡2(x)+1otherwise))−(5({1for im⁡(x)=0∧x mod π=01−tan⁡2(x)tan⁡2(x)+1otherwise))+8\left(- 5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 8
            //                                   /           /      pi\           \\
            ||            0               for And|im(x) = 0, |2*x + --| mod pi = 0||
        ___ ||                                   \           \      4 /           /|
8 - 5*\/ 2 *|<                                                                     |
            ||     2/    pi\    /    pi\                                           |
            ||2*sin |x + --|*cot|x + --|                  otherwise                |
            \\      \    8 /    \    8 /                                           /
(−52({0for im⁡(x)=0∧(2x+π4) mod π=02sin⁡2(x+π8)cot⁡(x+π8)otherwise))+8\left(- 5 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(2 x + \frac{\pi}{4}\right) \bmod \pi = 0 \\2 \sin^{2}{\left(x + \frac{\pi}{8} \right)} \cot{\left(x + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right)\right) + 8
      //      0        for And(im(x) = 0, 2*x mod pi = 0)\                                                  
      ||                                                 |     //   1      for And(im(x) = 0, x mod pi = 0)\
8 - 5*|<   /      pi\                                    | - 5*|<                                          |
      ||cos|2*x - --|              otherwise             |     \\cos(2*x)             otherwise            /
      \\   \      2 /                                    /
(−5({0for im⁡(x)=0∧2x mod π=0cos⁡(2x−π2)otherwise))−(5({1for im⁡(x)=0∧x mod π=0cos⁡(2x)otherwise))+8\left(- 5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\cos{\left(2 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\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}\right)\right) + 8
        ___    /      pi\
8 - 5*\/ 2 *cos|2*x - --|
               \      4 /
−52cos⁡(2x−π4)+8- 5 \sqrt{2} \cos{\left(2 x - \frac{\pi}{4} \right)} + 8
                                                          //      1        for And(im(x) = 0, x mod pi = 0)\
      //   0      for And(im(x) = 0, 2*x mod pi = 0)\     ||                                               |
      ||                                            |     ||      1                                        |
8 - 5*|<   1                                        | - 5*|<-------------             otherwise            |
      ||--------              otherwise             |     ||   /pi      \                                  |
      \\csc(2*x)                                    /     ||csc|-- - 2*x|                                  |
                                                          \\   \2       /                                  /
(−5({0for im⁡(x)=0∧2x mod π=01csc⁡(2x)otherwise))−(5({1for im⁡(x)=0∧x mod π=01csc⁡(−2x+π2)otherwise))+8\left(- 5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\frac{1}{\csc{\left(2 x \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + 8
      //      0        for And(im(x) = 0, 2*x mod pi = 0)\                                                  
      ||                                                 |     //   1      for And(im(x) = 0, x mod pi = 0)\
      ||      1                                          |     ||                                          |
8 - 5*|<-------------              otherwise             | - 5*|<   1                                      |
      ||   /      pi\                                    |     ||--------             otherwise            |
      ||sec|2*x - --|                                    |     \\sec(2*x)                                  /
      \\   \      2 /                                    /
(−5({0for im⁡(x)=0∧2x mod π=01sec⁡(2x−π2)otherwise))−(5({1for im⁡(x)=0∧x mod π=01sec⁡(2x)otherwise))+8\left(- 5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\sec{\left(2 x \right)}} & \text{otherwise} \end{cases}\right)\right) + 8
            //                         /           /      pi\           \\
            ||       0          for And|im(x) = 0, |2*x + --| mod pi = 0||
            ||                         \           \      4 /           /|
            ||                                                           |
        ___ ||      /    pi\                                             |
8 - 5*\/ 2 *|< 2*cot|x + --|                                             |
            ||      \    8 /                                             |
            ||----------------                  otherwise                |
            ||       2/    pi\                                           |
            ||1 + cot |x + --|                                           |
            \\        \    8 /                                           /
(−52({0for im⁡(x)=0∧(2x+π4) mod π=02cot⁡(x+π8)cot⁡2(x+π8)+1otherwise))+8\left(- 5 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(2 x + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(x + \frac{\pi}{8} \right)}}{\cot^{2}{\left(x + \frac{\pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 8
      //                       0                          for And(im(x) = 0, 2*x mod pi = 0)\     //                       1                         for And(im(x) = 0, x mod pi = 0)\
      ||                                                                                    |     ||                                                                                 |
      ||/     0       for And(im(x) = 0, 2*x mod pi = 0)                                    |     ||/     1        for And(im(x) = 0, x mod pi = 0)                                  |
      |||                                                                                   |     |||                                                                                |
8 - 5*|<|  2*cot(x)                                                                         | - 5*|<|        2                                                                       |
      ||<-----------              otherwise                           otherwise             |     ||<-1 + cot (x)                                               otherwise            |
      |||       2                                                                           |     |||------------             otherwise                                              |
      |||1 + cot (x)                                                                        |     |||       2                                                                        |
      \\\                                                                                   /     \\\1 + cot (x)                                                                     /
(−5({0for im⁡(x)=0∧2x mod π=0{0for im⁡(x)=0∧2x mod π=02cot⁡(x)cot⁡2(x)+1otherwiseotherwise))−(5({1for im⁡(x)=0∧x mod π=0{1for im⁡(x)=0∧x mod π=0cot⁡2(x)−1cot⁡2(x)+1otherwiseotherwise))+8\left(- 5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\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} & \text{otherwise} \end{cases}\right)\right) + 8
Комбинаторика [src]
8 - 5*cos(2*x) - 5*sin(2*x)
−5sin⁡(2x)−5cos⁡(2x)+8- 5 \sin{\left (2 x \right )} - 5 \cos{\left (2 x \right )} + 8
Раскрыть выражение [src]
           2                      
13 - 10*cos (x) - 10*cos(x)*sin(x)
−10sin⁡(x)cos⁡(x)−10cos⁡2(x)+13- 10 \sin{\left(x \right)} \cos{\left(x \right)} - 10 \cos^{2}{\left(x \right)} + 13