Дано
$$sqrt{12 sqrt{6}} left(frac{i}{12} sin{left (2 pi k + frac{7 pi}{4} right )} + frac{1}{12} cos{left (2 pi k + frac{7 pi}{4} right )}right)$$
Подстановка условия
$$sqrt{12 sqrt{6}} left(frac{i}{12} sin{left (2 pi k + frac{7 pi}{4} right )} + frac{1}{12} cos{left (2 pi k + frac{7 pi}{4} right )}right)$$
sqrt(12*sqrt(6))*(cos((7*pi)/4 + (2*pi)*k)/12 + ((3/2)*sin((7*pi)/4 + (2*pi)*k))/12)
$$sqrt{12 sqrt{6}} left(frac{(3/2)}{12} sin{left (2 pi k + frac{7 pi}{4} right )} + frac{1}{12} cos{left (2 pi k + frac{7 pi}{4} right )}right)$$
sqrt(12*sqrt(6))*(cos((7*pi)/4 + (2*pi)*k)/12 + (3*sin((7*pi)/4 + (2*pi)*k)/2)/12)
$$sqrt{12 sqrt{6}} left(frac{frac{3}{2}}{12} sin{left (2 pi k + frac{7 pi}{4} right )} + frac{1}{12} cos{left (2 pi k + frac{7 pi}{4} right )}right)$$
2*2^(1/4)*3^(3/4)*(-cos(pi/4 + 2*pi*k)/8 + sin(pi/4 + 2*pi*k)/12)
$$2 sqrt[4]{2} cdot 3^{frac{3}{4}} left(frac{1}{12} sin{left (2 pi k + frac{pi}{4} right )} – frac{1}{8} cos{left (2 pi k + frac{pi}{4} right )}right)$$
Степени
$$2 sqrt[4]{2} cdot 3^{frac{3}{4}} left(frac{1}{12} sin{left (2 pi k + frac{pi}{4} right )} – frac{i}{12} cos{left (2 pi k + frac{pi}{4} right )}right)$$
/ /pi /pi
|sin|– + 2*pi*k| I*cos|– + 2*pi*k||
4 ___ 3/4 | 4 / 4 /|
/ 2 *3 *|—————- – ——————|
6 6 /
$$sqrt[4]{2} cdot 3^{frac{3}{4}} left(frac{1}{6} sin{left (2 pi k + frac{pi}{4} right )} – frac{i}{6} cos{left (2 pi k + frac{pi}{4} right )}right)$$
Численный ответ
0.451801001804922*cos((7*pi)/4 + (2*pi)*k) + 0.451801001804922*i*sin((7*pi)/4 + (2*pi)*k)
Рациональный знаменатель
$$frac{sqrt[4]{2}}{6} 3^{frac{3}{4}} left(sin{left (2 pi k + frac{pi}{4} right )} – i cos{left (2 pi k + frac{pi}{4} right )}right)$$
Объединение рациональных выражений
$$frac{sqrt[4]{2}}{6} 3^{frac{3}{4}} left(i sin{left (pi left(2 k + frac{7}{4}right) right )} + cos{left (pi left(2 k + frac{7}{4}right) right )}right)$$
Общее упрощение
4 ____
/ 54 *(-I*cos(pi*(1/4 + 2*k)) + sin(pi*(1/4 + 2*k)))
—————————————————–
6
$$frac{sqrt[4]{54}}{6} left(sin{left (pi left(2 k + frac{1}{4}right) right )} – i cos{left (pi left(2 k + frac{1}{4}right) right )}right)$$
Соберем выражение
$$frac{sqrt[4]{2}}{6} 3^{frac{3}{4}} sin{left (2 pi k + frac{pi}{4} right )} – frac{sqrt[4]{2} i}{6} 3^{frac{3}{4}} cos{left (2 pi k + frac{pi}{4} right )}$$
/ /7*pi /7*pi
|I*sin|—- + 2*pi*k| cos|—- + 2*pi*k||
4 ___ 3/4 | 4 / 4 /|
2*/ 2 *3 *|——————– + ——————|
12 12 /
$$2 sqrt[4]{2} cdot 3^{frac{3}{4}} left(frac{i}{12} sin{left (2 pi k + frac{7 pi}{4} right )} + frac{1}{12} cos{left (2 pi k + frac{7 pi}{4} right )}right)$$
Общий знаменатель
4 ___ 3/4 /pi 4 ___ 3/4 /pi
/ 2 *3 *sin|– + 2*pi*k| I*/ 2 *3 *cos|– + 2*pi*k|
4 / 4 /
————————— – —————————–
6 6
$$frac{sqrt[4]{2}}{6} 3^{frac{3}{4}} sin{left (2 pi k + frac{pi}{4} right )} – frac{sqrt[4]{2} i}{6} 3^{frac{3}{4}} cos{left (2 pi k + frac{pi}{4} right )}$$
Тригонометрическая часть
4 ___ 3/4
/ 2 *3 *(-I*cos(pi*(1/4 + 2*k)) + sin(pi*(1/4 + 2*k)))
———————————————————
6
$$frac{sqrt[4]{2}}{6} 3^{frac{3}{4}} left(sin{left (pi left(2 k + frac{1}{4}right) right )} – i cos{left (pi left(2 k + frac{1}{4}right) right )}right)$$
Комбинаторика
4 ___ 3/4 / /pi /pi
-/ 2 *3 *|- sin|– + 2*pi*k| + I*cos|– + 2*pi*k||
4 / 4 //
——————————————————
6
$$- frac{sqrt[4]{2}}{6} 3^{frac{3}{4}} left(- sin{left (2 pi k + frac{pi}{4} right )} + i cos{left (2 pi k + frac{pi}{4} right )}right)$$
Раскрыть выражение
/ /7*pi /7*pi
|cos|—- + 2*pi*k| I*sin|—- + 2*pi*k||
4 ___ 3/4 | 4 / 4 /|
2*/ 2 *3 *|—————— + ——————–|
12 12 /
$$2 sqrt[4]{2} cdot 3^{frac{3}{4}} left(frac{i}{12} sin{left (2 pi k + frac{7 pi}{4} right )} + frac{1}{12} cos{left (2 pi k + frac{7 pi}{4} right )}right)$$
