Дано
$$frac{sin{left (4 a right )} + 1 – cos{left (4 a right )}}{cos{left (4 a right )} + 1 + sin{left (4 a right )}}$$
Подстановка условия
$$frac{sin{left (4 a right )} + 1 – cos{left (4 a right )}}{cos{left (4 a right )} + 1 + sin{left (4 a right )}}$$
(1 + sin(4*(1)) – cos(4*(1)))/(1 + cos(4*(1)) + sin(4*(1)))
$$frac{sin{left (4 (1) right )} + 1 – cos{left (4 (1) right )}}{cos{left (4 (1) right )} + 1 + sin{left (4 (1) right )}}$$
(1 + sin(4) – cos(4))/(1 + cos(4) + sin(4))
$$frac{sin{left (4 right )} + 1 – cos{left (4 right )}}{sin{left (4 right )} + cos{left (4 right )} + 1}$$
(1 – cos(4) + sin(4))/(1 + cos(4) + sin(4))
$$frac{sin{left (4 right )} – cos{left (4 right )} + 1}{sin{left (4 right )} + cos{left (4 right )} + 1}$$
Степени
$$frac{sin{left (4 a right )} – cos{left (4 a right )} + 1}{sin{left (4 a right )} + cos{left (4 a right )} + 1}$$
Численный ответ
(1.0 – cos(4*a) + sin(4*a))/(1.0 + cos(4*a) + sin(4*a))
Рациональный знаменатель
$$frac{sin{left (4 a right )} – cos{left (4 a right )} + 1}{sin{left (4 a right )} + cos{left (4 a right )} + 1}$$
Объединение рациональных выражений
$$frac{sin{left (4 a right )} – cos{left (4 a right )} + 1}{sin{left (4 a right )} + cos{left (4 a right )} + 1}$$
Общее упрощение
1 – cos(4*a) + sin(4*a)
———————–
___ / pi
1 + / 2 *sin|4*a + –|
4 /
$$frac{sin{left (4 a right )} – cos{left (4 a right )} + 1}{sqrt{2} sin{left (4 a + frac{pi}{4} right )} + 1}$$
Соберем выражение
$$frac{sin{left (4 a right )} – cos{left (4 a right )} + 1}{sin{left (4 a right )} + cos{left (4 a right )} + 1}$$
1 sin(4*a) cos(4*a)
———————– + ———————– – ———————–
1 + cos(4*a) + sin(4*a) 1 + cos(4*a) + sin(4*a) 1 + cos(4*a) + sin(4*a)
$$frac{sin{left (4 a right )}}{sin{left (4 a right )} + cos{left (4 a right )} + 1} – frac{cos{left (4 a right )}}{sin{left (4 a right )} + cos{left (4 a right )} + 1} + frac{1}{sin{left (4 a right )} + cos{left (4 a right )} + 1}$$
Общий знаменатель
2 + 2*sin(4*a)
-1 + ———————–
1 + cos(4*a) + sin(4*a)
$$frac{2 sin{left (4 a right )} + 2}{sin{left (4 a right )} + cos{left (4 a right )} + 1} – 1$$
Тригонометрическая часть
1 + sin(4*a) – cos(4*a)
———————–
___ / pi
1 + / 2 *sin|4*a + –|
4 /
$$frac{sin{left (4 a right )} + 1 – cos{left (4 a right )}}{sqrt{2} sin{left (4 a + frac{pi}{4} right )} + 1}$$
Комбинаторика
-(-1 – sin(4*a) + cos(4*a))
—————————-
1 + cos(4*a) + sin(4*a)
$$- frac{- sin{left (4 a right )} + cos{left (4 a right )} – 1}{sin{left (4 a right )} + cos{left (4 a right )} + 1}$$
Раскрыть выражение
4 4 3 3 2 2
1 – cos (a) – sin (a) – 4*sin (a)*cos(a) + 4*cos (a)*sin(a) + 6*cos (a)*sin (a)
——————————————————————————-
4 4 2 2 3 3
1 + cos (a) + sin (a) – 6*cos (a)*sin (a) – 4*sin (a)*cos(a) + 4*cos (a)*sin(a)
$$frac{- sin^{4}{left (a right )} – 4 sin^{3}{left (a right )} cos{left (a right )} + 6 sin^{2}{left (a right )} cos^{2}{left (a right )} + 4 sin{left (a right )} cos^{3}{left (a right )} – cos^{4}{left (a right )} + 1}{sin^{4}{left (a right )} – 4 sin^{3}{left (a right )} cos{left (a right )} – 6 sin^{2}{left (a right )} cos^{2}{left (a right )} + 4 sin{left (a right )} cos^{3}{left (a right )} + cos^{4}{left (a right )} + 1}$$
