Господин Экзамен

Другие калькуляторы

Упрощение выражений/ Общий знаменатель/ cos(pi/7+a)*cos(5*pi/14-a)-sin(pi/7+a)*sin(5*pi/14)

Общий знаменатель cos(pi/7+a)*cos(5*pi/14-a)-sin(pi/7+a)*sin(5*pi/14)

Выражение, которое надо упростить:

Решение

Вы ввели [src] 
   /pi    \    /5*pi    \      /pi    \    /5*pi\
cos|-- + a|*cos|---- - a| - sin|-- + a|*sin|----|
   \7     /    \ 14     /      \7     /    \ 14 /
$$\cos{\left(- a + \frac{5 \pi}{14} \right)} \cos{\left(a + \frac{\pi}{7} \right)} - \sin{\left(\frac{5 \pi}{14} \right)} \sin{\left(a + \frac{\pi}{7} \right)}$$ 
cos(pi/7 + a)*cos(5*pi/14 - a) - sin(pi/7 + a)*sin(5*pi/14)
Общее упрощение [src] 
/     /pi\      /    pi\\    /    pi\
|- cos|--| + cos|a + --||*sin|a + --|
\     \7 /      \    7 //    \    7 /
$$\left(\cos{\left(a + \frac{\pi}{7} \right)} - \cos{\left(\frac{\pi}{7} \right)}\right) \sin{\left(a + \frac{\pi}{7} \right)}$$ 
(-cos(pi/7) + cos(a + pi/7))*sin(a + pi/7)
Собрать выражение [src] 
   /      2*pi\               /    2*pi\
sin|2*a + ----|            sin|a + ----|
   \       7  /   sin(a)      \     7  /
--------------- - ------ - -------------
       2            2            2
$$- \frac{\sin{\left(a \right)}}{2} - \frac{\sin{\left(a + \frac{2 \pi}{7} \right)}}{2} + \frac{\sin{\left(2 a + \frac{2 \pi}{7} \right)}}{2}$$ 
sin(2*a + 2*pi/7)/2 - sin(a)/2 - sin(a + 2*pi/7)/2
Рациональный знаменатель [src] 
/     /5*pi\      /    pi\\    /    pi\
|- sin|----| + cos|a + --||*sin|a + --|
\     \ 14 /      \    7 //    \    7 /
$$\left(\cos{\left(a + \frac{\pi}{7} \right)} - \sin{\left(\frac{5 \pi}{14} \right)}\right) \sin{\left(a + \frac{\pi}{7} \right)}$$ 
   /    pi\    /    pi\      /5*pi\    /    pi\
cos|a + --|*sin|a + --| - sin|----|*sin|a + --|
   \    7 /    \    7 /      \ 14 /    \    7 /
$$\sin{\left(a + \frac{\pi}{7} \right)} \cos{\left(a + \frac{\pi}{7} \right)} - \sin{\left(\frac{5 \pi}{14} \right)} \sin{\left(a + \frac{\pi}{7} \right)}$$ 
cos(a + pi/7)*sin(a + pi/7) - sin(5*pi/14)*sin(a + pi/7)
Объединение рациональных выражений [src] 
   /pi + 7*a\    /-14*a + 5*pi\      /pi + 7*a\    /5*pi\
cos|--------|*cos|------------| - sin|--------|*sin|----|
   \   7    /    \     14     /      \   7    /    \ 14 /
$$\cos{\left(\frac{- 14 a + 5 \pi}{14} \right)} \cos{\left(\frac{7 a + \pi}{7} \right)} - \sin{\left(\frac{5 \pi}{14} \right)} \sin{\left(\frac{7 a + \pi}{7} \right)}$$ 
cos((pi + 7*a)/7)*cos((-14*a + 5*pi)/14) - sin((pi + 7*a)/7)*sin(5*pi/14)
Степени [src] 
   /    pi\    /    pi\      /5*pi\    /    pi\
cos|a + --|*sin|a + --| - sin|----|*sin|a + --|
   \    7 /    \    7 /      \ 14 /    \    7 /
$$\sin{\left(a + \frac{\pi}{7} \right)} \cos{\left(a + \frac{\pi}{7} \right)} - \sin{\left(\frac{5 \pi}{14} \right)} \sin{\left(a + \frac{\pi}{7} \right)}$$ 
/   /    5*pi\      /     5*pi\\ /   /    pi\      /     pi\\   /     /     pi\      /    pi\\ /   -5*pi*I    5*pi*I\
| I*|a - ----|    I*|-a + ----|| | I*|a + --|    I*|-a - --||   |   I*|-a - --|    I*|a + --|| |   -------    ------|
|   \     14 /      \      14 /| |   \    7 /      \     7 /|   |     \     7 /      \    7 /| |      14        14  |
|e               e             | |e             e           |   \- e            + e          /*\- e        + e      /
|------------- + --------------|*|----------- + ------------| + -----------------------------------------------------
\      2               2       / \     2             2      /                             4
$$\left(\frac{e^{i \left(- a - \frac{\pi}{7}\right)}}{2} + \frac{e^{i \left(a + \frac{\pi}{7}\right)}}{2}\right) \left(\frac{e^{i \left(- a + \frac{5 \pi}{14}\right)}}{2} + \frac{e^{i \left(a - \frac{5 \pi}{14}\right)}}{2}\right) + \frac{\left(- e^{- \frac{5 i \pi}{14}} + e^{\frac{5 i \pi}{14}}\right) \left(- e^{i \left(- a - \frac{\pi}{7}\right)} + e^{i \left(a + \frac{\pi}{7}\right)}\right)}{4}$$ 
(exp(i*(a - 5*pi/14))/2 + exp(i*(-a + 5*pi/14))/2)*(exp(i*(a + pi/7))/2 + exp(i*(-a - pi/7))/2) + (-exp(i*(-a - pi/7)) + exp(i*(a + pi/7)))*(-exp(-5*pi*i/14) + exp(5*pi*i/14))/4
Комбинаторика [src] 
/     /5*pi\      /    pi\\    /    pi\
|- sin|----| + cos|a + --||*sin|a + --|
\     \ 14 /      \    7 //    \    7 /
$$\left(\cos{\left(a + \frac{\pi}{7} \right)} - \sin{\left(\frac{5 \pi}{14} \right)}\right) \sin{\left(a + \frac{\pi}{7} \right)}$$ 
(-sin(5*pi/14) + cos(a + pi/7))*sin(a + pi/7)
Общий знаменатель [src] 
   /    pi\    /    pi\      /5*pi\    /    pi\
cos|a + --|*sin|a + --| - sin|----|*sin|a + --|
   \    7 /    \    7 /      \ 14 /    \    7 /
$$\sin{\left(a + \frac{\pi}{7} \right)} \cos{\left(a + \frac{\pi}{7} \right)} - \sin{\left(\frac{5 \pi}{14} \right)} \sin{\left(a + \frac{\pi}{7} \right)}$$ 
cos(a + pi/7)*sin(a + pi/7) - sin(5*pi/14)*sin(a + pi/7)
Раскрыть выражение [src] 
/          /pi\             /pi\\ /          /5*pi\             /5*pi\\   /          /pi\      /pi\       \    /5*pi\
|cos(a)*cos|--| - sin(a)*sin|--||*|cos(a)*cos|----| + sin(a)*sin|----|| - |cos(a)*sin|--| + cos|--|*sin(a)|*sin|----|
\          \7 /             \7 // \          \ 14 /             \ 14 //   \          \7 /      \7 /       /    \ 14 /
$$\left(- \sin{\left(\frac{\pi}{7} \right)} \sin{\left(a \right)} + \cos{\left(\frac{\pi}{7} \right)} \cos{\left(a \right)}\right) \left(\sin{\left(\frac{5 \pi}{14} \right)} \sin{\left(a \right)} + \cos{\left(\frac{5 \pi}{14} \right)} \cos{\left(a \right)}\right) - \left(\sin{\left(a \right)} \cos{\left(\frac{\pi}{7} \right)} + \sin{\left(\frac{\pi}{7} \right)} \cos{\left(a \right)}\right) \sin{\left(\frac{5 \pi}{14} \right)}$$ 
                                          _______________                                                                            _______________                         _______________               
                                         /        /2*pi\                                                       /2*pi\               /        /2*pi\                         /        /2*pi\                
                                        /      cos|----|                                             cos(a)*cos|----|*sin(a)       /      cos|----|                        /      cos|----|                
     2/pi\          cos(a)*sin(a)      /   1      \ 7  /     2       /pi\      2/pi\                           \ 7  /             /   1      \ 7  /     2       /pi\      /   1      \ 7  /            /pi\
- cos |--|*sin(a) - ------------- +   /    - - --------- *cos (a)*cos|--| + cos |--|*cos(a)*sin(a) + ----------------------- -   /    - - --------- *sin (a)*cos|--| -   /    - - --------- *cos(a)*cos|--|
      \7 /                2         \/     2       2                 \7 /       \7 /                            2              \/     2       2                 \7 /   \/     2       2                \7 /
$$- \sqrt{- \frac{\cos{\left(\frac{2 \pi}{7} \right)}}{2} + \frac{1}{2}} \sin^{2}{\left(a \right)} \cos{\left(\frac{\pi}{7} \right)} - \frac{\sin{\left(a \right)} \cos{\left(a \right)}}{2} + \frac{\sin{\left(a \right)} \cos{\left(\frac{2 \pi}{7} \right)} \cos{\left(a \right)}}{2} + \sin{\left(a \right)} \cos^{2}{\left(\frac{\pi}{7} \right)} \cos{\left(a \right)} + \sqrt{- \frac{\cos{\left(\frac{2 \pi}{7} \right)}}{2} + \frac{1}{2}} \cos{\left(\frac{\pi}{7} \right)} \cos^{2}{\left(a \right)} - \sin{\left(a \right)} \cos^{2}{\left(\frac{\pi}{7} \right)} - \sqrt{- \frac{\cos{\left(\frac{2 \pi}{7} \right)}}{2} + \frac{1}{2}} \cos{\left(\frac{\pi}{7} \right)} \cos{\left(a \right)}$$ 
-cos(pi/7)^2*sin(a) - cos(a)*sin(a)/2 + sqrt(1/2 - cos(2*pi/7)/2)*cos(a)^2*cos(pi/7) + cos(pi/7)^2*cos(a)*sin(a) + cos(a)*cos(2*pi/7)*sin(a)/2 - sqrt(1/2 - cos(2*pi/7)/2)*sin(a)^2*cos(pi/7) - sqrt(1/2 - cos(2*pi/7)/2)*cos(a)*cos(pi/7)
Численный ответ [src] 
-0.900968867902419*sin(pi/7 + a) + cos(pi/7 + a)*cos(5*pi/14 - a)
-0.900968867902419*sin(pi/7 + a) + cos(pi/7 + a)*cos(5*pi/14 - a)
    © Господин Экзамен – Калькулятор