Калькуляторы онлайн/ Решение уравнений/ Упрощение выражений/ Общий знаменатель/ 2*((sin(x)^3/(1+cos(x))^3+(6*sin(x))/(1+cos(x)))/24-(cos(x)+1)/(8*sin(x)))

Общий знаменатель 2*((sin(x)^3/(1+cos(x))^3 ... 24-(cos(x)+1)/(8*sin(x)))

Учитель очень удивится увидев твоё верное решение 😼

Примеры

Решение

Вы ввели [src]
  /      3                                \
  |   sin (x)       6*sin(x)              |
  |------------- + ----------             |
  |            3   1 + cos(x)             |
  |(1 + cos(x))                 cos(x) + 1|
2*|-------------------------- - ----------|
  \            24                8*sin(x) /
$$2 \left(\frac{1}{24} \left(\frac{\sin^{3}{\left (x \right )}}{\left(\cos{\left (x \right )} + 1\right)^{3}} + \frac{6 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}\right) - \left(\cos{\left (x \right )} + 1\right) \frac{1}{8 \sin{\left (x \right )}}\right)$$
Степени [src]
                                     3        
    sin(x)       1 + cos(x)       sin (x)     
-------------- - ---------- + ----------------
2*(1 + cos(x))    4*sin(x)                   3
                              12*(1 + cos(x))
$$- \frac{\cos{\left (x \right )} + 1}{4 \sin{\left (x \right )}} + \frac{\sin{\left (x \right )}}{2 \cos{\left (x \right )} + 2} + \frac{\sin^{3}{\left (x \right )}}{12 \left(\cos{\left (x \right )} + 1\right)^{3}}$$
                   /  1   cos(x)\                   
                 2*|- - - ------|          3        
    sin(x)         \  8     8   /       sin (x)     
-------------- + ---------------- + ----------------
2*(1 + cos(x))        sin(x)                       3
                                    12*(1 + cos(x))
$$\frac{- \frac{1}{4} \cos{\left (x \right )} - \frac{1}{4}}{\sin{\left (x \right )}} + \frac{\sin{\left (x \right )}}{2 \cos{\left (x \right )} + 2} + \frac{\sin^{3}{\left (x \right )}}{12 \left(\cos{\left (x \right )} + 1\right)^{3}}$$
Численный ответ [src]
0.5*sin(x)/(1.0 + cos(x)) + 0.0833333333333333*sin(x)^3/(1.0 + cos(x))^3 - 0.25*(1.0 + cos(x))/sin(x)
Рациональный знаменатель [src]
   /   3                                 3       \                         4              
16*\sin (x)*(1 + cos(x)) + 6*(1 + cos(x)) *sin(x)/*sin(x) + 48*(1 + cos(x)) *(-1 - cos(x))
------------------------------------------------------------------------------------------
                                                 4                                        
                                 192*(1 + cos(x)) *sin(x)
$$\frac{1}{192 \left(\cos{\left (x \right )} + 1\right)^{4} \sin{\left (x \right )}} \left(16 \left(6 \left(\cos{\left (x \right )} + 1\right)^{3} \sin{\left (x \right )} + \left(\cos{\left (x \right )} + 1\right) \sin^{3}{\left (x \right )}\right) \sin{\left (x \right )} + 48 \left(- \cos{\left (x \right )} - 1\right) \left(\cos{\left (x \right )} + 1\right)^{4}\right)$$
Объединение рациональных выражений [src]
                4      2    /   2                    2\
- 3*(1 + cos(x))  + sin (x)*\sin (x) + 6*(1 + cos(x)) /
-------------------------------------------------------
                               3                       
                12*(1 + cos(x)) *sin(x)
$$\frac{1}{12 \left(\cos{\left (x \right )} + 1\right)^{3} \sin{\left (x \right )}} \left(\left(6 \left(\cos{\left (x \right )} + 1\right)^{2} + \sin^{2}{\left (x \right )}\right) \sin^{2}{\left (x \right )} - 3 \left(\cos{\left (x \right )} + 1\right)^{4}\right)$$
Общее упрощение [src]
   4                    4                 2    2   
sin (x) - 3*(1 + cos(x))  + 6*(1 + cos(x)) *sin (x)
---------------------------------------------------
                             3                     
              12*(1 + cos(x)) *sin(x)
$$\frac{1}{12 \left(\cos{\left (x \right )} + 1\right)^{3} \sin{\left (x \right )}} \left(- 3 \left(\cos{\left (x \right )} + 1\right)^{4} + 6 \left(\cos{\left (x \right )} + 1\right)^{2} \sin^{2}{\left (x \right )} + \sin^{4}{\left (x \right )}\right)$$
Собрать выражение [src]
                     /      3                   \
                     |   sin (x)       6*sin(x) |
                   2*|------------- + ----------|
                     |            3   1 + cos(x)|
  2*(cos(x) + 1)     \(1 + cos(x))              /
- -------------- + ------------------------------
     8*sin(x)                    24
$$\frac{1}{24} \left(\frac{2 \sin^{3}{\left (x \right )}}{\left(\cos{\left (x \right )} + 1\right)^{3}} + \frac{12 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}\right) - 2 \left(\cos{\left (x \right )} + 1\right) \frac{1}{8 \sin{\left (x \right )}}$$
                            660*sin(3*x)                                                      202*sin(2*x)                                                      96*sin(4*x)                                                       64*sin(4*x)                                                        60*sin(x)                                                        60*sin(5*x)                                                        6*sin(6*x)                                                        48*sin(x)                                                        102*sin(2*x)                                                      136*sin(3*x)                         
- --------------------------------------------------------------- - --------------------------------------------------------------- - --------------------------------------------------------------- - --------------------------------------------------------------- - --------------------------------------------------------------- - --------------------------------------------------------------- - --------------------------------------------------------------- + --------------------------------------------------------------- + --------------------------------------------------------------- + ---------------------------------------------------------------
  3360 + 96*cos(4*x) + 768*cos(3*x) + 2688*cos(2*x) + 5376*cos(x)   1680 + 48*cos(4*x) + 384*cos(3*x) + 1344*cos(2*x) + 2688*cos(x)   3360 + 96*cos(4*x) + 768*cos(3*x) + 2688*cos(2*x) + 5376*cos(x)   1680 + 48*cos(4*x) + 384*cos(3*x) + 1344*cos(2*x) + 2688*cos(x)   1680 + 48*cos(4*x) + 384*cos(3*x) + 1344*cos(2*x) + 2688*cos(x)   3360 + 96*cos(4*x) + 768*cos(3*x) + 2688*cos(2*x) + 5376*cos(x)   3360 + 96*cos(4*x) + 768*cos(3*x) + 2688*cos(2*x) + 5376*cos(x)   3360 + 96*cos(4*x) + 768*cos(3*x) + 2688*cos(2*x) + 5376*cos(x)   3360 + 96*cos(4*x) + 768*cos(3*x) + 2688*cos(2*x) + 5376*cos(x)   1680 + 48*cos(4*x) + 384*cos(3*x) + 1344*cos(2*x) + 2688*cos(x)
$$\frac{48 \sin{\left (x \right )}}{5376 \cos{\left (x \right )} + 2688 \cos{\left (2 x \right )} + 768 \cos{\left (3 x \right )} + 96 \cos{\left (4 x \right )} + 3360} + \frac{102 \sin{\left (2 x \right )}}{5376 \cos{\left (x \right )} + 2688 \cos{\left (2 x \right )} + 768 \cos{\left (3 x \right )} + 96 \cos{\left (4 x \right )} + 3360} - \frac{660 \sin{\left (3 x \right )}}{5376 \cos{\left (x \right )} + 2688 \cos{\left (2 x \right )} + 768 \cos{\left (3 x \right )} + 96 \cos{\left (4 x \right )} + 3360} - \frac{96 \sin{\left (4 x \right )}}{5376 \cos{\left (x \right )} + 2688 \cos{\left (2 x \right )} + 768 \cos{\left (3 x \right )} + 96 \cos{\left (4 x \right )} + 3360} - \frac{60 \sin{\left (5 x \right )}}{5376 \cos{\left (x \right )} + 2688 \cos{\left (2 x \right )} + 768 \cos{\left (3 x \right )} + 96 \cos{\left (4 x \right )} + 3360} - \frac{6 \sin{\left (6 x \right )}}{5376 \cos{\left (x \right )} + 2688 \cos{\left (2 x \right )} + 768 \cos{\left (3 x \right )} + 96 \cos{\left (4 x \right )} + 3360} - \frac{60 \sin{\left (x \right )}}{2688 \cos{\left (x \right )} + 1344 \cos{\left (2 x \right )} + 384 \cos{\left (3 x \right )} + 48 \cos{\left (4 x \right )} + 1680} - \frac{202 \sin{\left (2 x \right )}}{2688 \cos{\left (x \right )} + 1344 \cos{\left (2 x \right )} + 384 \cos{\left (3 x \right )} + 48 \cos{\left (4 x \right )} + 1680} + \frac{136 \sin{\left (3 x \right )}}{2688 \cos{\left (x \right )} + 1344 \cos{\left (2 x \right )} + 384 \cos{\left (3 x \right )} + 48 \cos{\left (4 x \right )} + 1680} - \frac{64 \sin{\left (4 x \right )}}{2688 \cos{\left (x \right )} + 1344 \cos{\left (2 x \right )} + 384 \cos{\left (3 x \right )} + 48 \cos{\left (4 x \right )} + 1680}$$
Общий знаменатель [src]
 /       4           2           4            3                        2            2                  2       2   \ 
-\3 - sin (x) - 6*sin (x) + 3*cos (x) + 12*cos (x) + 12*cos(x) + 18*cos (x) - 12*sin (x)*cos(x) - 6*cos (x)*sin (x)/ 
---------------------------------------------------------------------------------------------------------------------
                                           3                   2                                                     
                         12*sin(x) + 12*cos (x)*sin(x) + 36*cos (x)*sin(x) + 36*cos(x)*sin(x)
$$- \frac{- \sin^{4}{\left (x \right )} - 6 \sin^{2}{\left (x \right )} \cos^{2}{\left (x \right )} - 12 \sin^{2}{\left (x \right )} \cos{\left (x \right )} - 6 \sin^{2}{\left (x \right )} + 3 \cos^{4}{\left (x \right )} + 12 \cos^{3}{\left (x \right )} + 18 \cos^{2}{\left (x \right )} + 12 \cos{\left (x \right )} + 3}{12 \sin{\left (x \right )} \cos^{3}{\left (x \right )} + 36 \sin{\left (x \right )} \cos^{2}{\left (x \right )} + 36 \sin{\left (x \right )} \cos{\left (x \right )} + 12 \sin{\left (x \right )}}$$
Тригонометрическая часть [src]
                                     3        
    sin(x)       cos(x) + 1       sin (x)     
-------------- - ---------- + ----------------
2*(1 + cos(x))    4*sin(x)                   3
                              12*(1 + cos(x))
$$- \frac{\cos{\left (x \right )} + 1}{4 \sin{\left (x \right )}} + \frac{\sin{\left (x \right )}}{2 \cos{\left (x \right )} + 2} + \frac{\sin^{3}{\left (x \right )}}{12 \left(\cos{\left (x \right )} + 1\right)^{3}}$$
Комбинаторика [src]
 /       4           2           4            3                        2            2                  2       2   \ 
-\3 - sin (x) - 6*sin (x) + 3*cos (x) + 12*cos (x) + 12*cos(x) + 18*cos (x) - 12*sin (x)*cos(x) - 6*cos (x)*sin (x)/ 
---------------------------------------------------------------------------------------------------------------------
                                                              3                                                      
                                               12*(1 + cos(x)) *sin(x)
$$- \frac{1}{12 \left(\cos{\left (x \right )} + 1\right)^{3} \sin{\left (x \right )}} \left(- \sin^{4}{\left (x \right )} - 6 \sin^{2}{\left (x \right )} \cos^{2}{\left (x \right )} - 12 \sin^{2}{\left (x \right )} \cos{\left (x \right )} - 6 \sin^{2}{\left (x \right )} + 3 \cos^{4}{\left (x \right )} + 12 \cos^{3}{\left (x \right )} + 18 \cos^{2}{\left (x \right )} + 12 \cos{\left (x \right )} + 3\right)$$
Раскрыть выражение [src]
  /      3                   \             
  |   sin (x)       6*sin(x) |             
2*|------------- + ----------|             
  |            3   1 + cos(x)|             
  \(1 + cos(x))              /   cos(x) + 1
------------------------------ - ----------
              24                  4*sin(x)
$$\frac{1}{24} \left(\frac{2 \sin^{3}{\left (x \right )}}{\left(\cos{\left (x \right )} + 1\right)^{3}} + \frac{12 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}\right) - \frac{\cos{\left (x \right )} + 1}{4 \sin{\left (x \right )}}$$
                                              3        
    sin(x)       (1 + cos(x))*sin(x)       sin (x)     
-------------- - ------------------- + ----------------
2*(1 + cos(x))            4                           3
                                       12*(1 + cos(x))
$$- \frac{1}{4} \left(\cos{\left (x \right )} + 1\right) \sin{\left (x \right )} + \frac{\sin{\left (x \right )}}{2 \cos{\left (x \right )} + 2} + \frac{\sin^{3}{\left (x \right )}}{12 \left(\cos{\left (x \right )} + 1\right)^{3}}$$