Sin п 2 х 1 2 – Найдите корни уравнения sin(2x-п/2)=-1/2 принадлежащие…

Найдите корни уравнения sin(2x-п/2)=-1/2 принадлежащие…

Ответ оставил Гость

По формуле приведения через π/2 меняет sin на cos. но синус в 4 четверти отрицателен.
 
 
 Отбор корней на промежутке (0;3π/2)
 Для корня x = π/6 + πn,
Если 
Если 

Для корня 
Если 

Оцени ответ

shkolniku.com

(1+sin(3*p*1/2+2*x)-sin(x)^2)*1/sin(2*x)*sin(p*1/2+x)+sin(x)*cos(p-2*x) если p=-1/4 (упростите выражение)

Решение

       /3*p      \      2                                    
1 + sin|--- + 2*x| - sin (x)                                 
       \ 2       /              /p    \                      
----------------------------*sin|- + x| + sin(x)*cos(p - 2*x)
          sin(2*x)              \2    /                      

$$\frac{1}{\sin{\left (2 x \right )}} \left(\sin{\left (\frac{3 p}{2} + 2 x \right )} + 1 — \sin^{2}{\left (x \right )}\right) \sin{\left (\frac{p}{2} + x \right )} + \sin{\left (x \right )} \cos{\left (p — 2 x \right )}$$

Подстановка условия

[LaTeX]

((1 + sin((3*p)/2 + 2*x) - sin(x)^2)/sin(2*x))*sin(p/2 + x) + sin(x)*cos(p - 2*x) при p = -1/4
((1 + sin((3*p)/2 + 2*x) - sin(x)^2)/sin(2*x))*sin(p/2 + x) + sin(x)*cos(p - 2*x)

$$\frac{1}{\sin{\left (2 x \right )}} \left(\sin{\left (\frac{3 p}{2} + 2 x \right )} + 1 — \sin^{2}{\left (x \right )}\right) \sin{\left (\frac{p}{2} + x \right )} + \sin{\left (x \right )} \cos{\left (p — 2 x \right )}$$

((1 + sin((3*(-1/4))/2 + 2*x) - sin(x)^2)/sin(2*x))*sin((-1/4)/2 + x) + sin(x)*cos((-1/4) - 2*x)

$$\frac{1}{\sin{\left (2 x \right )}} \left(\sin{\left (\frac{3 (-1/4)}{2} + 2 x \right )} + 1 — \sin^{2}{\left (x \right )}\right) \sin{\left (\frac{(-1/4)}{2} + x \right )} + \sin{\left (x \right )} \cos{\left ((-1/4) — 2 x \right )}$$

((1 + sin((3*(-1)/4)/2 + 2*x) - sin(x)^2)/sin(2*x))*sin(-1/8 + x) + sin(x)*cos(-1/4 - 2*x)

$$\frac{1}{\sin{\left (2 x \right )}} \left(\sin{\left (2 x + \frac{- \frac{3}{4}}{2} 1 \right )} + 1 — \sin^{2}{\left (x \right )}\right) \sin{\left (x — \frac{1}{8} \right )} + \sin{\left (x \right )} \cos{\left (- 2 x — \frac{1}{4} \right )}$$

cos(1/4 + 2*x)*sin(x) + (1 - sin(x)^2 + sin(-3/8 + 2*x))*sin(-1/8 + x)/sin(2*x)

$$\frac{\sin{\left (x — \frac{1}{8} \right )}}{\sin{\left (2 x \right )}} \left(- \sin^{2}{\left (x \right )} + \sin{\left (2 x — \frac{3}{8} \right )} + 1\right) + \sin{\left (x \right )} \cos{\left (2 x + \frac{1}{4} \right )}$$

                      /       2         /      3*p\\    /    p\
                      |1 - sin (x) + sin|2*x + ---||*sin|x + -|
                      \                 \       2 //    \    2/
cos(p - 2*x)*sin(x) + -----------------------------------------
                                       sin(2*x)                

$$\frac{\sin{\left (\frac{p}{2} + x \right )}}{\sin{\left (2 x \right )}} \left(- \sin^{2}{\left (x \right )} + \sin{\left (\frac{3 p}{2} + 2 x \right )} + 1\right) + \sin{\left (x \right )} \cos{\left (p — 2 x \right )}$$

Численный ответ

[LaTeX]

cos(p - 2*x)*sin(x) + (1.0 - sin(x)^2 + sin((3*p)/2 + 2*x))*sin(p/2 + x)/sin(2*x)
Рациональный знаменатель

[LaTeX]

   /    p\    /      3*p\      2       /    p\                                     /    p\
sin|x + -|*sin|2*x + ---| - sin (x)*sin|x + -| + cos(p - 2*x)*sin(x)*sin(2*x) + sin|x + -|
   \    2/    \       2 /              \    2/                                     \    2/
------------------------------------------------------------------------------------------
                                         sin(2*x)                                         

$$\frac{1}{\sin{\left (2 x \right )}} \left(- \sin^{2}{\left (x \right )} \sin{\left (\frac{p}{2} + x \right )} + \sin{\left (x \right )} \sin{\left (2 x \right )} \cos{\left (p — 2 x \right )} + \sin{\left (\frac{p}{2} + x \right )} \sin{\left (\frac{3 p}{2} + 2 x \right )} + \sin{\left (\frac{p}{2} + x \right )}\right)$$

Объединение рациональных выражений

[LaTeX]

/       2         /3*p + 4*x\\    /p + 2*x\                               
|1 - sin (x) + sin|---------||*sin|-------| + cos(p - 2*x)*sin(x)*sin(2*x)
\                 \    2    //    \   2   /                               
--------------------------------------------------------------------------
                                 sin(2*x)                                 

$$\frac{1}{\sin{\left (2 x \right )}} \left(\left(- \sin^{2}{\left (x \right )} + \sin{\left (\frac{1}{2} \left(3 p + 4 x\right) \right )} + 1\right) \sin{\left (\frac{1}{2} \left(p + 2 x\right) \right )} + \sin{\left (x \right )} \sin{\left (2 x \right )} \cos{\left (p — 2 x \right )}\right)$$

Общее упрощение

[LaTeX]

/       2         /      3*p\\    /    p\                               
|1 - sin (x) + sin|2*x + ---||*sin|x + -| + cos(p - 2*x)*sin(x)*sin(2*x)
\                 \       2 //    \    2/                               
------------------------------------------------------------------------
                                sin(2*x)                                

$$\frac{1}{\sin{\left (2 x \right )}} \left(\left(- \sin^{2}{\left (x \right )} + \sin{\left (\frac{3 p}{2} + 2 x \right )} + 1\right) \sin{\left (\frac{p}{2} + x \right )} + \sin{\left (x \right )} \sin{\left (2 x \right )} \cos{\left (p — 2 x \right )}\right)$$

Собрать выражение

[LaTeX]

                            /                /    p\                       /p    \      /p      \\         
                            |             sin|x + -|                    sin|- - x|   sin|- + 3*x||         
sin(p - x)   sin(p - 3*x)   |cos(p + x)      \    2/   cos(2*p + 3*x)      \2    /      \2      /|         
---------- - ------------ + |---------- + ---------- - -------------- + ---------- + ------------|*csc(2*x)
    2             2         \    2            2              2              4             4      /         

$$\left(\frac{1}{4} \sin{\left (\frac{p}{2} — x \right )} + \frac{1}{2} \sin{\left (\frac{p}{2} + x \right )} + \frac{1}{4} \sin{\left (\frac{p}{2} + 3 x \right )} + \frac{1}{2} \cos{\left (p + x \right )} — \frac{1}{2} \cos{\left (2 p + 3 x \right )}\right) \csc{\left (2 x \right )} — \frac{1}{2} \sin{\left (p — 3 x \right )} + \frac{1}{2} \sin{\left (p — x \right )}$$

Комбинаторика

[LaTeX]

   /    p\    /      3*p\      2       /    p\                                     /    p\
sin|x + -|*sin|2*x + ---| - sin (x)*sin|x + -| + cos(p - 2*x)*sin(x)*sin(2*x) + sin|x + -|
   \    2/    \       2 /              \    2/                                     \    2/
------------------------------------------------------------------------------------------
                                         sin(2*x)                                         

$$\frac{1}{\sin{\left (2 x \right )}} \left(- \sin^{2}{\left (x \right )} \sin{\left (\frac{p}{2} + x \right )} + \sin{\left (x \right )} \sin{\left (2 x \right )} \cos{\left (p — 2 x \right )} + \sin{\left (\frac{p}{2} + x \right )} \sin{\left (\frac{3 p}{2} + 2 x \right )} + \sin{\left (\frac{p}{2} + x \right )}\right)$$

Общий знаменатель

[LaTeX]

   /    p\    /      3*p\      2       /    p\      /    p\                      
sin|x + -|*sin|2*x + ---| - sin (x)*sin|x + -| + sin|x + -|                      
   \    2/    \       2 /              \    2/      \    2/                      
----------------------------------------------------------- + cos(p - 2*x)*sin(x)
                          sin(2*x)                                               

$$\frac{1}{\sin{\left (2 x \right )}} \left(- \sin^{2}{\left (x \right )} \sin{\left (\frac{p}{2} + x \right )} + \sin{\left (\frac{p}{2} + x \right )} \sin{\left (\frac{3 p}{2} + 2 x \right )} + \sin{\left (\frac{p}{2} + x \right )}\right) + \sin{\left (x \right )} \cos{\left (p — 2 x \right )}$$

Тригонометрическая часть

[LaTeX]

                      /       2         /      3*p\\    /p    \
                      |1 - sin (x) + sin|2*x + ---||*sin|- + x|
                      \                 \       2 //    \2    /
cos(p - 2*x)*sin(x) + -----------------------------------------
                                       sin(2*x)                

$$\frac{\sin{\left (\frac{p}{2} + x \right )}}{\sin{\left (2 x \right )}} \left(- \sin^{2}{\left (x \right )} + \sin{\left (\frac{3 p}{2} + 2 x \right )} + 1\right) + \sin{\left (x \right )} \cos{\left (p — 2 x \right )}$$

Раскрыть выражение

[LaTeX]

                                             /          /p\      /p\       \ /       2                  /3*p\      /3*p\         \
                                             |cos(x)*sin|-| + cos|-|*sin(x)|*|1 - sin (x) + cos(2*x)*sin|---| + cos|---|*sin(2*x)|
                                             \          \2/      \2/       / \                          \ 2 /      \ 2 /         /
(cos(p)*cos(2*x) + sin(p)*sin(2*x))*sin(x) + -------------------------------------------------------------------------------------
                                                                                2*cos(x)*sin(x)                                   

$$\frac{1}{2 \sin{\left (x \right )} \cos{\left (x \right )}} \left(\sin{\left (\frac{p}{2} \right )} \cos{\left (x \right )} + \sin{\left (x \right )} \cos{\left (\frac{p}{2} \right )}\right) \left(\sin{\left (\frac{3 p}{2} \right )} \cos{\left (2 x \right )} — \sin^{2}{\left (x \right )} + \sin{\left (2 x \right )} \cos{\left (\frac{3 p}{2} \right )} + 1\right) + \left(\sin{\left (p \right )} \sin{\left (2 x \right )} + \cos{\left (p \right )} \cos{\left (2 x \right )}\right) \sin{\left (x \right )}$$

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(1+sin(3*p*1/2+2*x)-sin(x)^2)*1/(sin(2*x)*sin(p*1/2+x)+sin(x)*cos(p-2*x)) если p=3 (упростите выражение)

Решение

              /3*p      \      2         
       1 + sin|--- + 2*x| - sin (x)      
              \ 2       /                
-----------------------------------------
            /p    \                      
sin(2*x)*sin|- + x| + sin(x)*cos(p - 2*x)
            \2    /                      

$$\frac{\sin{\left (\frac{3 p}{2} + 2 x \right )} + 1 — \sin^{2}{\left (x \right )}}{\sin{\left (x \right )} \cos{\left (p — 2 x \right )} + \sin{\left (2 x \right )} \sin{\left (\frac{p}{2} + x \right )}}$$

Подстановка условия

[LaTeX]

(1 + sin((3*p)/2 + 2*x) - sin(x)^2)/(sin(2*x)*sin(p/2 + x) + sin(x)*cos(p - 2*x)) при p = 3
(1 + sin((3*p)/2 + 2*x) - sin(x)^2)/(sin(2*x)*sin(p/2 + x) + sin(x)*cos(p - 2*x))

$$\frac{\sin{\left (\frac{3 p}{2} + 2 x \right )} + 1 — \sin^{2}{\left (x \right )}}{\sin{\left (x \right )} \cos{\left (p — 2 x \right )} + \sin{\left (2 x \right )} \sin{\left (\frac{p}{2} + x \right )}}$$

(1 + sin((3*(3))/2 + 2*x) - sin(x)^2)/(sin(2*x)*sin((3)/2 + x) + sin(x)*cos((3) - 2*x))

$$\frac{\sin{\left (\frac{3 (3)}{2} + 2 x \right )} + 1 — \sin^{2}{\left (x \right )}}{\sin{\left (x \right )} \cos{\left ((3) — 2 x \right )} + \sin{\left (2 x \right )} \sin{\left (\frac{(3)}{2} + x \right )}}$$

(1 + sin((3*3)/2 + 2*x) - sin(x)^2)/(sin(2*x)*sin(3/2 + x) + sin(x)*cos(3 - 2*x))

$$\frac{\sin{\left (2 x + \frac{9}{2} 1 \right )} + 1 — \sin^{2}{\left (x \right )}}{\sin{\left (x \right )} \cos{\left (- 2 x + 3 \right )} + \sin{\left (2 x \right )} \sin{\left (x + \frac{3}{2} \right )}}$$

(1 - sin(x)^2 + sin(9/2 + 2*x))/(cos(-3 + 2*x)*sin(x) + sin(2*x)*sin(3/2 + x))

$$\frac{- \sin^{2}{\left (x \right )} + \sin{\left (2 x + \frac{9}{2} \right )} + 1}{\sin{\left (x \right )} \cos{\left (2 x — 3 \right )} + \sin{\left (2 x \right )} \sin{\left (x + \frac{3}{2} \right )}}$$

              2         /      3*p\      
       1 - sin (x) + sin|2*x + ---|      
                        \       2 /      
-----------------------------------------
                                  /    p\
cos(p - 2*x)*sin(x) + sin(2*x)*sin|x + -|
                                  \    2/

$$\frac{- \sin^{2}{\left (x \right )} + \sin{\left (\frac{3 p}{2} + 2 x \right )} + 1}{\sin{\left (x \right )} \cos{\left (p — 2 x \right )} + \sin{\left (2 x \right )} \sin{\left (\frac{p}{2} + x \right )}}$$

Численный ответ

[LaTeX]

(1.0 - sin(x)^2 + sin((3*p)/2 + 2*x))/(cos(p - 2*x)*sin(x) + sin(2*x)*sin(p/2 + x))
Рациональный знаменатель

[LaTeX]

              2         /      3*p\      
       1 - sin (x) + sin|2*x + ---|      
                        \       2 /      
-----------------------------------------
                                  /    p\
cos(p - 2*x)*sin(x) + sin(2*x)*sin|x + -|
                                  \    2/

$$\frac{- \sin^{2}{\left (x \right )} + \sin{\left (\frac{3 p}{2} + 2 x \right )} + 1}{\sin{\left (x \right )} \cos{\left (p — 2 x \right )} + \sin{\left (2 x \right )} \sin{\left (\frac{p}{2} + x \right )}}$$

Объединение рациональных выражений

[LaTeX]

               2         /3*p + 4*x\       
        1 - sin (x) + sin|---------|       
                         \    2    /       
-------------------------------------------
                         /p + 2*x\         
cos(p - 2*x)*sin(x) + sin|-------|*sin(2*x)
                         \   2   /         

$$\frac{- \sin^{2}{\left (x \right )} + \sin{\left (\frac{1}{2} \left(3 p + 4 x\right) \right )} + 1}{\sin{\left (x \right )} \cos{\left (p — 2 x \right )} + \sin{\left (2 x \right )} \sin{\left (\frac{1}{2} \left(p + 2 x\right) \right )}}$$

Общее упрощение

[LaTeX]

              2         /      3*p\      
       1 - sin (x) + sin|2*x + ---|      
                        \       2 /      
-----------------------------------------
                                  /    p\
cos(p - 2*x)*sin(x) + sin(2*x)*sin|x + -|
                                  \    2/

$$\frac{- \sin^{2}{\left (x \right )} + \sin{\left (\frac{3 p}{2} + 2 x \right )} + 1}{\sin{\left (x \right )} \cos{\left (p — 2 x \right )} + \sin{\left (2 x \right )} \sin{\left (\frac{p}{2} + x \right )}}$$

Собрать выражение

[LaTeX]

                                                                                                                                               /      3*p\                   
                                                                                                                                          2*sin|2*x + ---|                   
                             1                                                      cos(2*x)                                                   \       2 /                   
- ------------------------------------------------------- - ------------------------------------------------------- - -------------------------------------------------------
       /p    \                   /p      \                       /p    \                   /p      \                       /p    \                   /p      \               
  - cos|- - x| - sin(p - x) + cos|- + 3*x| + sin(p - 3*x)   - cos|- - x| - sin(p - x) + cos|- + 3*x| + sin(p - 3*x)   - cos|- - x| - sin(p - x) + cos|- + 3*x| + sin(p - 3*x)
       \2    /                   \2      /                       \2    /                   \2      /                       \2    /                   \2      /               

$$- \frac{2 \sin{\left (\frac{3 p}{2} + 2 x \right )}}{\sin{\left (p — 3 x \right )} — \sin{\left (p — x \right )} — \cos{\left (\frac{p}{2} — x \right )} + \cos{\left (\frac{p}{2} + 3 x \right )}} — \frac{\cos{\left (2 x \right )}}{\sin{\left (p — 3 x \right )} — \sin{\left (p — x \right )} — \cos{\left (\frac{p}{2} — x \right )} + \cos{\left (\frac{p}{2} + 3 x \right )}} — \frac{1}{\sin{\left (p — 3 x \right )} — \sin{\left (p — x \right )} — \cos{\left (\frac{p}{2} — x \right )} + \cos{\left (\frac{p}{2} + 3 x \right )}}$$

              2         /3*p      \      
       1 - sin (x) + sin|--- + 2*x|      
                        \ 2       /      
-----------------------------------------
                                  /p    \
sin(x)*cos(p - 2*x) + sin(2*x)*sin|- + x|
                                  \2    /

$$\frac{- \sin^{2}{\left (x \right )} + \sin{\left (\frac{3 p}{2} + 2 x \right )} + 1}{\sin{\left (x \right )} \cos{\left (p — 2 x \right )} + \sin{\left (2 x \right )} \sin{\left (\frac{p}{2} + x \right )}}$$

Общий знаменатель

[LaTeX]

              2         /      3*p\      
       1 - sin (x) + sin|2*x + ---|      
                        \       2 /      
-----------------------------------------
                                  /    p\
cos(p - 2*x)*sin(x) + sin(2*x)*sin|x + -|
                                  \    2/

$$\frac{- \sin^{2}{\left (x \right )} + \sin{\left (\frac{3 p}{2} + 2 x \right )} + 1}{\sin{\left (x \right )} \cos{\left (p — 2 x \right )} + \sin{\left (2 x \right )} \sin{\left (\frac{p}{2} + x \right )}}$$

Тригонометрическая часть

[LaTeX]

              2         /      3*p\      
       1 - sin (x) + sin|2*x + ---|      
                        \       2 /      
-----------------------------------------
                                  /p    \
cos(p - 2*x)*sin(x) + sin(2*x)*sin|- + x|
                                  \2    /

$$\frac{- \sin^{2}{\left (x \right )} + \sin{\left (\frac{3 p}{2} + 2 x \right )} + 1}{\sin{\left (x \right )} \cos{\left (p — 2 x \right )} + \sin{\left (2 x \right )} \sin{\left (\frac{p}{2} + x \right )}}$$

Комбинаторика

[LaTeX]

              2         /      3*p\      
       1 - sin (x) + sin|2*x + ---|      
                        \       2 /      
-----------------------------------------
                                  /    p\
cos(p - 2*x)*sin(x) + sin(2*x)*sin|x + -|
                                  \    2/

$$\frac{- \sin^{2}{\left (x \right )} + \sin{\left (\frac{3 p}{2} + 2 x \right )} + 1}{\sin{\left (x \right )} \cos{\left (p — 2 x \right )} + \sin{\left (2 x \right )} \sin{\left (\frac{p}{2} + x \right )}}$$

Раскрыть выражение

[LaTeX]

                           2                  /3*p\      /3*p\                              
                    1 - sin (x) + cos(2*x)*sin|---| + cos|---|*sin(2*x)                     
                                              \ 2 /      \ 2 /                              
--------------------------------------------------------------------------------------------
                                               /          /p\      /p\       \              
(cos(p)*cos(2*x) + sin(p)*sin(2*x))*sin(x) + 2*|cos(x)*sin|-| + cos|-|*sin(x)|*cos(x)*sin(x)
                                               \          \2/      \2/       /              

$$\frac{\sin{\left (\frac{3 p}{2} \right )} \cos{\left (2 x \right )} — \sin^{2}{\left (x \right )} + \sin{\left (2 x \right )} \cos{\left (\frac{3 p}{2} \right )} + 1}{2 \left(\sin{\left (\frac{p}{2} \right )} \cos{\left (x \right )} + \sin{\left (x \right )} \cos{\left (\frac{p}{2} \right )}\right) \sin{\left (x \right )} \cos{\left (x \right )} + \left(\sin{\left (p \right )} \sin{\left (2 x \right )} + \cos{\left (p \right )} \cos{\left (2 x \right )}\right) \sin{\left (x \right )}}$$

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