Для начала запишем разложение углов в сумму двух углов:
sin(7π/12) = sin(π/3 + π/4) = sin(π/3) cos(π/4) + cos(π/3) sin(π/4) = sqrt(3)/2 sqrt(2)/2 + 1/2 sqrt(2)/2 = sqrt(6)/4 + sqrt(2)/4 = (sqrt(6) + sqrt(2))/4
sin(π/12) = sin(π/3 - π/4) = sin(π/3) cos(π/4) - cos(π/3) sin(π/4) = sqrt(3)/2 sqrt(2)/2 - 1/2 sqrt(2)/2 = sqrt(6)/4 - sqrt(2)/4 = (sqrt(6) - sqrt(2))/4
cos(π/12) = cos(π/3 - π/4) = cos(π/3) cos(π/4) + sin(π/3) sin(π/4) = 1/2 sqrt(2)/2 + sqrt(3)/2 sqrt(2)/2 = sqrt(2)/4 + sqrt(6)/4 = (sqrt(2) + sqrt(6))/4
cos(7π/12) = cos(π/3 + π/4) = cos(π/3) cos(π/4) - sin(π/3) sin(π/4) = 1/2 sqrt(2)/2 - sqrt(3)/2 sqrt(2)/2 = sqrt(2)/4 - sqrt(6)/4 = (sqrt(2) - sqrt(6))/4
Теперь можем подставить все полученные значения в исходное выражение:
(sin(7π/12) + sin(π/12) + cos(π/12) - cos(7π/12)) = ((sqrt(6) + sqrt(2))/4 + (sqrt(6) - sqrt(2))/4 + (sqrt(2) + sqrt(6))/4 - (sqrt(2) - sqrt(6))/4) = 2 sqrt(6) / 4 + 2 sqrt(6) / 4 = sqrt(6)
Ответ: sqrt(6)
Для начала запишем разложение углов в сумму двух углов:
sin(7π/12) = sin(π/3 + π/4) = sin(π/3) cos(π/4) + cos(π/3) sin(π/4) = sqrt(3)/2 sqrt(2)/2 + 1/2 sqrt(2)/2 = sqrt(6)/4 + sqrt(2)/4 = (sqrt(6) + sqrt(2))/4
sin(π/12) = sin(π/3 - π/4) = sin(π/3) cos(π/4) - cos(π/3) sin(π/4) = sqrt(3)/2 sqrt(2)/2 - 1/2 sqrt(2)/2 = sqrt(6)/4 - sqrt(2)/4 = (sqrt(6) - sqrt(2))/4
cos(π/12) = cos(π/3 - π/4) = cos(π/3) cos(π/4) + sin(π/3) sin(π/4) = 1/2 sqrt(2)/2 + sqrt(3)/2 sqrt(2)/2 = sqrt(2)/4 + sqrt(6)/4 = (sqrt(2) + sqrt(6))/4
cos(7π/12) = cos(π/3 + π/4) = cos(π/3) cos(π/4) - sin(π/3) sin(π/4) = 1/2 sqrt(2)/2 - sqrt(3)/2 sqrt(2)/2 = sqrt(2)/4 - sqrt(6)/4 = (sqrt(2) - sqrt(6))/4
Теперь можем подставить все полученные значения в исходное выражение:
(sin(7π/12) + sin(π/12) + cos(π/12) - cos(7π/12)) = ((sqrt(6) + sqrt(2))/4 + (sqrt(6) - sqrt(2))/4 + (sqrt(2) + sqrt(6))/4 - (sqrt(2) - sqrt(6))/4) = 2 sqrt(6) / 4 + 2 sqrt(6) / 4 = sqrt(6)
Ответ: sqrt(6)