cos(2*a)+sin(2*a)-5еслиa=4 (упростите выражение)

Вы ввели

cos(2*a) + sin(2*a) - 5

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

Подстановка условия

cos(2*a) + sin(2*a) - 1*5 при a = 4

подставляем

cos(2*a) + sin(2*a) - 5

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

       ___    /      pi\
-5 + \/ 2 *sin|2*a + --|
              \      4 /

$$\sqrt{2} \sin{\left(2 a + \frac{\pi}{4} \right)} - 5$$

переменные

a = 4

$$a = 4$$

       ___    /        pi\
-5 + \/ 2 *sin|2*(4) + --|
              \        4 /

$$\sqrt{2} \sin{\left(2 (4) + \frac{\pi}{4} \right)} - 5$$

       ___    /      pi\
-5 + \/ 2 *sin|2*4 + --|
              \      4 /

$$-5 + \sqrt{2} \sin{\left(\frac{\pi}{4} + 2 \cdot 4 \right)}$$

       ___    /    pi\
-5 + \/ 2 *sin|8 + --|
              \    4 /

$$-5 + \sqrt{2} \sin{\left(\frac{\pi}{4} + 8 \right)}$$

Степени

      -2*I*a    2*I*a     /   -2*I*a    2*I*a\
     e         e        I*\- e       + e     /
-5 + ------- + ------ - ----------------------
        2        2                2           

$$- \frac{i \left(e^{2 i a} - e^{- 2 i a}\right)}{2} + \frac{e^{2 i a}}{2} - 5 + \frac{e^{- 2 i a}}{2}$$

Численный ответ

-5.0 + cos(2*a) + sin(2*a)

Общее упрощение

       ___    /      pi\
-5 + \/ 2 *sin|2*a + --|
              \      4 /

$$\sqrt{2} \sin{\left(2 a + \frac{\pi}{4} \right)} - 5$$

Тригонометрическая часть

           //                         /           /      pi\           \\
           ||       0          for And|im(a) = 0, |2*a + --| mod pi = 0||
           ||                         \           \      4 /           /|
           ||                                                           |
       ___ ||      /    pi\                                             |
-5 + \/ 2 *|< 2*cot|a + --|                                             |
           ||      \    8 /                                             |
           ||----------------                  otherwise                |
           ||       2/    pi\                                           |
           ||1 + cot |a + --|                                           |
           \\        \    8 /                                           /

$$\left(\sqrt{2} \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(2 a + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(a + \frac{\pi}{8} \right)}}{\cot^{2}{\left(a + \frac{\pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - 5$$

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

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

                   /pi      \
-5 + sin(2*a) + sin|-- + 2*a|
                   \2       /

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

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

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

                   /      pi\
-5 + cos(2*a) + cos|2*a - --|
                   \      2 /

$$\cos{\left(2 a \right)} + \cos{\left(2 a - \frac{\pi}{2} \right)} - 5$$

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

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

            2                 
     1 - tan (a)     2*tan(a) 
-5 + ----------- + -----------
            2             2   
     1 + tan (a)   1 + tan (a)

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

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

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

         ___    /    pi\
     2*\/ 2 *tan|a + --|
                \    8 /
-5 + -------------------
              2/    pi\ 
       1 + tan |a + --| 
               \    8 / 

$$-5 + \frac{2 \sqrt{2} \tan{\left(a + \frac{\pi}{8} \right)}}{\tan^{2}{\left(a + \frac{\pi}{8} \right)} + 1}$$

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

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

        1             1      
-5 + -------- + -------------
     sec(2*a)      /      pi\
                sec|2*a - --|
                   \      2 /

$$-5 + \frac{1}{\sec{\left(2 a - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(2 a \right)}}$$

           ___    
         \/ 2     
-5 + -------------
        /      pi\
     csc|2*a + --|
        \      4 /

$$-5 + \frac{\sqrt{2}}{\csc{\left(2 a + \frac{\pi}{4} \right)}}$$

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

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

           ___    
         \/ 2     
-5 + -------------
        /      pi\
     sec|2*a - --|
        \      4 /

$$-5 + \frac{\sqrt{2}}{\sec{\left(2 a - \frac{\pi}{4} \right)}}$$

       ___    /      pi\
-5 + \/ 2 *sin|2*a + --|
              \      4 /

$$\sqrt{2} \sin{\left(2 a + \frac{\pi}{4} \right)} - 5$$

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

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

       ___    /      pi\
-5 + \/ 2 *cos|2*a - --|
              \      4 /

$$\sqrt{2} \cos{\left(2 a - \frac{\pi}{4} \right)} - 5$$

        1             1      
-5 + -------- + -------------
     csc(2*a)      /pi      \
                csc|-- - 2*a|
                   \2       /

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

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

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

           //                                   /           /      pi\           \\
           ||            0               for And|im(a) = 0, |2*a + --| mod pi = 0||
       ___ ||                                   \           \      4 /           /|
-5 + \/ 2 *|<                                                                     |
           ||     2/    pi\    /    pi\                                           |
           ||2*sin |a + --|*cot|a + --|                  otherwise                |
           \\      \    8 /    \    8 /                                           /

$$\left(\sqrt{2} \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(2 a + \frac{\pi}{4}\right) \bmod \pi = 0 \\2 \sin^{2}{\left(a + \frac{\pi}{8} \right)} \cot{\left(a + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right)\right) - 5$$

     //                      0                        for And(im(a) = 0, 2*a mod pi = 0)\   //                     1                       for And(im(a) = 0, a mod pi = 0)\
     ||                                                                                 |   ||                                                                             |
-5 + |

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

        1          1    
-5 + -------- + --------
     csc(2*a)   sec(2*a)

$$-5 + \frac{1}{\sec{\left(2 a \right)}} + \frac{1}{\csc{\left(2 a \right)}}$$

Раскрыть выражение

          2                     
-6 + 2*cos (a) + 2*cos(a)*sin(a)

$$2 \sin{\left(a \right)} \cos{\left(a \right)} + 2 \cos^{2}{\left(a \right)} - 6$$