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