Дано
$$frac{1}{384} left(8 cos{left (6 x right )} + 3 cos{left (8 x right )} – 12 cos{left (4 x right )} – 72 cos{left (2 x right )}right)$$
Подстановка условия
$$frac{1}{384} left(8 cos{left (6 x right )} + 3 cos{left (8 x right )} – 12 cos{left (4 x right )} – 72 cos{left (2 x right )}right)$$
(3*cos(8*(2)) + 8*cos(6*(2)) – 12*cos(4*(2)) – 72*cos(2*(2)))/384
$$frac{1}{384} left(8 cos{left (6 (2) right )} + 3 cos{left (8 (2) right )} – 12 cos{left (4 (2) right )} – 72 cos{left (2 (2) right )}right)$$
(3*cos(8*2) + 8*cos(6*2) – 12*cos(4*2) – 72*cos(2*2))/384
$$frac{1}{384} left(- 12 cos{left (2 cdot 4 right )} + 3 cos{left (2 cdot 8 right )} + 8 cos{left (2 cdot 6 right )} – 72 cos{left (2 cdot 2 right )}right)$$
-3*cos(4)/16 – cos(8)/32 + cos(12)/48 + cos(16)/128
$$frac{1}{128} cos{left (16 right )} – frac{1}{32} cos{left (8 right )} + frac{1}{48} cos{left (12 right )} – frac{3}{16} cos{left (4 right )}$$
Степени
$$- frac{3}{16} cos{left (2 x right )} – frac{1}{32} cos{left (4 x right )} + frac{1}{48} cos{left (6 x right )} + frac{1}{128} cos{left (8 x right )}$$
Численный ответ
0.0078125*cos(8*x) + 0.0208333333333333*cos(6*x) – 0.03125*cos(4*x) – 0.1875*cos(2*x)
Рациональный знаменатель
$$- frac{3}{16} cos{left (2 x right )} – frac{1}{32} cos{left (4 x right )} + frac{1}{48} cos{left (6 x right )} + frac{1}{128} cos{left (8 x right )}$$
Объединение рациональных выражений
$$- frac{3}{16} cos{left (2 x right )} – frac{1}{32} cos{left (4 x right )} + frac{1}{48} cos{left (6 x right )} + frac{1}{128} cos{left (8 x right )}$$
Общее упрощение
3*cos(2*x) cos(4*x) cos(6*x) cos(8*x)
– ———- – ——– + ——– + ——–
16 32 48 128
$$- frac{3}{16} cos{left (2 x right )} – frac{1}{32} cos{left (4 x right )} + frac{1}{48} cos{left (6 x right )} + frac{1}{128} cos{left (8 x right )}$$
Соберем выражение
$$- frac{3}{16} cos{left (2 x right )} – frac{1}{32} cos{left (4 x right )} + frac{1}{48} cos{left (6 x right )} + frac{1}{128} cos{left (8 x right )}$$
cos(6*x) cos(8*x) 12*cos(4*x) 72*cos(2*x)
——– + ——– – ———– – ———–
48 128 384 384
$$- frac{3}{16} cos{left (2 x right )} – frac{1}{32} cos{left (4 x right )} + frac{1}{48} cos{left (6 x right )} + frac{1}{128} cos{left (8 x right )}$$
Общий знаменатель
3*cos(2*x) cos(4*x) cos(6*x) cos(8*x)
– ———- – ——– + ——– + ——–
16 32 48 128
$$- frac{3}{16} cos{left (2 x right )} – frac{1}{32} cos{left (4 x right )} + frac{1}{48} cos{left (6 x right )} + frac{1}{128} cos{left (8 x right )}$$
Тригонометрическая часть
3*cos(2*x) cos(4*x) cos(6*x) cos(8*x)
– ———- – ——– + ——– + ——–
16 32 48 128
$$- frac{3}{16} cos{left (2 x right )} – frac{1}{32} cos{left (4 x right )} + frac{1}{48} cos{left (6 x right )} + frac{1}{128} cos{left (8 x right )}$$
Комбинаторика
-(-8*cos(6*x) – 3*cos(8*x) + 12*cos(4*x) + 72*cos(2*x))
——————————————————–
384
$$- frac{1}{384} left(72 cos{left (2 x right )} + 12 cos{left (4 x right )} – 8 cos{left (6 x right )} – 3 cos{left (8 x right )}right)$$
