1+sqrt(2)*cos(a)еслиa=-1/3 (упростите выражение)

Вы ввели

      ___       
1 + \/ 2 *cos(a)

$$\sqrt{2} \cos{\left(a \right)} + 1$$

Подстановка условия

1 + sqrt(2)*cos(a) при a = -1/3

подставляем

      ___       
1 + \/ 2 *cos(a)

$$\sqrt{2} \cos{\left(a \right)} + 1$$

      ___       
1 + \/ 2 *cos(a)

$$\sqrt{2} \cos{\left(a \right)} + 1$$

переменные

a = -1/3

$$a = - \frac{1}{3}$$

      ___            
1 + \/ 2 *cos((-1/3))

$$\sqrt{2} \cos{\left((-1/3) \right)} + 1$$

      ___          
1 + \/ 2 *cos(-1/3)

$$1 + \sqrt{2} \cos{\left(- \frac{1}{3} \right)}$$

      ___         
1 + \/ 2 *cos(1/3)

$$1 + \sqrt{2} \cos{\left(\frac{1}{3} \right)}$$

Степени

          / I*a    -I*a\
      ___ |e      e    |
1 + \/ 2 *|---- + -----|
          \ 2       2  /

$$\sqrt{2} \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right) + 1$$

Численный ответ

1.0 + 1.4142135623731*cos(a)

Тригонометрическая часть

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

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

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

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

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

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

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

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

      ___ 
    \/ 2  
1 + ------
    sec(a)

$$1 + \frac{\sqrt{2}}{\sec{\left(a \right)}}$$

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

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

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

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

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

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

      ___ /       2/a\\
    \/ 2 *|1 - tan |-||
          \        \2//
1 + -------------------
               2/a\    
        1 + tan |-|    
                \2/    

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

         ___   
       \/ 2    
1 + -----------
       /pi    \
    csc|-- - a|
       \2     /

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

      ___    /    pi\
1 + \/ 2 *sin|a + --|
             \    2 /

$$\sqrt{2} \sin{\left(a + \frac{\pi}{2} \right)} + 1$$

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

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