Калькуляторы онлайн/ Решение уравнений/ Упрощение выражений/ sin(6*atan(x))/192+3*sin(4*atan(x))/64+15*sin(2*atan(x))/64 если x=2

sin(6*atan(x))/192+3*sin( ... in(2*atan(x))/64 если x=2 (упростите выражение)

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Примеры

Решение

Вы ввели [src]
sin(6*atan(x))   3*sin(4*atan(x))   15*sin(2*atan(x))
-------------- + ---------------- + -----------------
     192                64                  64
$$\frac{3}{64} \sin{\left (4 \operatorname{atan}{\left (x \right )} \right )} + \frac{1}{192} \sin{\left (6 \operatorname{atan}{\left (x \right )} \right )} + \frac{15}{64} \sin{\left (2 \operatorname{atan}{\left (x \right )} \right )}$$
Подстановка условия [src]
sin(6*atan(x))/192 + (3*sin(4*atan(x)))/64 + (15*sin(2*atan(x)))/64 при x = 2
sin(6*atan(x))/192 + (3*sin(4*atan(x)))/64 + (15*sin(2*atan(x)))/64
$$\frac{3}{64} \sin{\left (4 \operatorname{atan}{\left (x \right )} \right )} + \frac{1}{192} \sin{\left (6 \operatorname{atan}{\left (x \right )} \right )} + \frac{15}{64} \sin{\left (2 \operatorname{atan}{\left (x \right )} \right )}$$
sin(6*atan((2)))/192 + (3*sin(4*atan((2))))/64 + (15*sin(2*atan((2))))/64
$$\frac{3}{64} \sin{\left (4 \operatorname{atan}{\left ((2) \right )} \right )} + \frac{1}{192} \sin{\left (6 \operatorname{atan}{\left ((2) \right )} \right )} + \frac{15}{64} \sin{\left (2 \operatorname{atan}{\left ((2) \right )} \right )}$$
sin(6*atan(2))/192 + (3*sin(4*atan(2)))/64 + (15*sin(2*atan(2)))/64
$$\frac{3}{64} \sin{\left (4 \operatorname{atan}{\left (2 \right )} \right )} + \frac{1}{192} \sin{\left (6 \operatorname{atan}{\left (2 \right )} \right )} + \frac{15}{64} \sin{\left (2 \operatorname{atan}{\left (2 \right )} \right )}$$
sin(6*atan(2))/192 + 3*sin(4*atan(2))/64 + 15*sin(2*atan(2))/64
$$\frac{3}{64} \sin{\left (4 \operatorname{atan}{\left (2 \right )} \right )} + \frac{1}{192} \sin{\left (6 \operatorname{atan}{\left (2 \right )} \right )} + \frac{15}{64} \sin{\left (2 \operatorname{atan}{\left (2 \right )} \right )}$$
Степени [src]
sin(6*atan(x))   3*sin(4*atan(x))   15*sin(2*atan(x))
-------------- + ---------------- + -----------------
     192                64                  64
$$\frac{15}{64} \sin{\left (2 \operatorname{atan}{\left (x \right )} \right )} + \frac{3}{64} \sin{\left (4 \operatorname{atan}{\left (x \right )} \right )} + \frac{1}{192} \sin{\left (6 \operatorname{atan}{\left (x \right )} \right )}$$
Численный ответ [src]
0.046875*sin(4*atan(x)) + 0.234375*sin(2*atan(x)) + 0.00520833333333333*sin(6*atan(x))
Рациональный знаменатель [src]
sin(6*atan(x))   3*sin(4*atan(x))   15*sin(2*atan(x))
-------------- + ---------------- + -----------------
     192                64                  64
$$\frac{15}{64} \sin{\left (2 \operatorname{atan}{\left (x \right )} \right )} + \frac{3}{64} \sin{\left (4 \operatorname{atan}{\left (x \right )} \right )} + \frac{1}{192} \sin{\left (6 \operatorname{atan}{\left (x \right )} \right )}$$
Объединение рациональных выражений [src]
9*sin(4*atan(x)) + 45*sin(2*atan(x)) + sin(6*atan(x))
-----------------------------------------------------
                         192
$$\frac{1}{192} \left(45 \sin{\left (2 \operatorname{atan}{\left (x \right )} \right )} + 9 \sin{\left (4 \operatorname{atan}{\left (x \right )} \right )} + \sin{\left (6 \operatorname{atan}{\left (x \right )} \right )}\right)$$
Общее упрощение [src]
                                   /     2\
sin(6*atan(x))       15*x      3*x*\1 - x /
-------------- + ----------- + ------------
     192            /     2\              2
                 32*\1 + x /      /     2\ 
                               16*\1 + x /
$$\frac{3 x \left(- x^{2} + 1\right)}{16 \left(x^{2} + 1\right)^{2}} + \frac{15 x}{32 x^{2} + 32} + \frac{1}{192} \sin{\left (6 \operatorname{atan}{\left (x \right )} \right )}$$
Собрать выражение [src]
3*sin(4*atan(x))   15*sin(2*atan(x))   sin(6*atan(x))
---------------- + ----------------- + --------------
       64                  64               192
$$\frac{15}{64} \sin{\left (2 \operatorname{atan}{\left (x \right )} \right )} + \frac{3}{64} \sin{\left (4 \operatorname{atan}{\left (x \right )} \right )} + \frac{1}{192} \sin{\left (6 \operatorname{atan}{\left (x \right )} \right )}$$
sin(6*atan(x))   3*sin(4*atan(x))   15*sin(2*atan(x))
-------------- + ---------------- + -----------------
     192                64                  64
$$\frac{15}{64} \sin{\left (2 \operatorname{atan}{\left (x \right )} \right )} + \frac{3}{64} \sin{\left (4 \operatorname{atan}{\left (x \right )} \right )} + \frac{1}{192} \sin{\left (6 \operatorname{atan}{\left (x \right )} \right )}$$
Общий знаменатель [src]
sin(6*atan(x))   3*sin(4*atan(x))   15*sin(2*atan(x))
-------------- + ---------------- + -----------------
     192                64                  64
$$\frac{15}{64} \sin{\left (2 \operatorname{atan}{\left (x \right )} \right )} + \frac{3}{64} \sin{\left (4 \operatorname{atan}{\left (x \right )} \right )} + \frac{1}{192} \sin{\left (6 \operatorname{atan}{\left (x \right )} \right )}$$
Тригонометрическая часть [src]
                                   /     2\
sin(6*atan(x))       15*x      3*x*\1 - x /
-------------- + ----------- + ------------
     192            /     2\              2
                 32*\1 + x /      /     2\ 
                               16*\1 + x /
$$\frac{3 x \left(- x^{2} + 1\right)}{16 \left(x^{2} + 1\right)^{2}} + \frac{15 x}{32 x^{2} + 32} + \frac{1}{192} \sin{\left (6 \operatorname{atan}{\left (x \right )} \right )}$$
Комбинаторика [src]
9*sin(4*atan(x)) + 45*sin(2*atan(x)) + sin(6*atan(x))
-----------------------------------------------------
                         192
$$\frac{1}{192} \left(45 \sin{\left (2 \operatorname{atan}{\left (x \right )} \right )} + 9 \sin{\left (4 \operatorname{atan}{\left (x \right )} \right )} + \sin{\left (6 \operatorname{atan}{\left (x \right )} \right )}\right)$$
Раскрыть выражение [src]
                    3              3                            5                    
   15*x          5*x            3*x             x              x             3*x     
---------- - ------------ - ------------ + ------------ + ------------ + ------------
         2              3              2              3              3              2
32 + 32*x       /     2\       /     2\       /     2\       /     2\       /     2\ 
             48*\1 + x /    16*\1 + x /    32*\1 + x /    32*\1 + x /    16*\1 + x /
$$\frac{x^{5}}{32 \left(x^{2} + 1\right)^{3}} - \frac{3 x^{3}}{16 \left(x^{2} + 1\right)^{2}} - \frac{5 x^{3}}{48 \left(x^{2} + 1\right)^{3}} + \frac{15 x}{32 x^{2} + 32} + \frac{3 x}{16 \left(x^{2} + 1\right)^{2}} + \frac{x}{32 \left(x^{2} + 1\right)^{3}}$$