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