Для начала разберемся с умножением тригонометрических функций:
cos(α)sin(β) = (1/2) * [sin(α + β) - sin(α - β)]
sin(α)sin(β) = (1/2) * [cos(α - β) - cos(α + β)]
Теперь применим эти формулы:
cos 170 sin35 - cos35 sin 170 sin105 sin75 + sin 15 cos 105
= (1/2) [sin(170 + 35) - sin(170 - 35)] - (1/2) [cos(35 - 170) - cos(35 + 170)] * sin105 sin75 + sin 15 cos 105
= (1/2) [sin 205 - sin 135] - (1/2) [-cos 135 - cos 205] (1/2) [cos(105 - 75) - cos(105 + 75)] + (1/2) * [sin(15 + 105) + sin(105 - 15)]
= (1/2) [sin 205 - sin 135] - (1/2) [cos 135 + cos 205] (1/2) [cos 30 - cos 180] + (1/2) * [sin 120 + sin 90]
= (1/2) [sin 205 - sin 135] - (1/2) [cos 135 + cos 205] (1/2) [sqrt(3)/2 - (-1/2)] + (1/2) * [sqrt(3)/2 + 1]
= (1/2) [sin 205 - sin 135] - (1/4) [sqrt(3)(cos 135 + cos 205) + cos 135 + cos 205] + (1/4) * sqrt(3) + 1
= (1/2) [sin 205 - sin 135] - (1/4) [sqrt(3)(-√2/2 + -√2/2) + -√2/2 + √2/2] + (1/4) * sqrt(3) + 1
= (1/2) [-√2/2 - -√2/2] - (1/4) [-√3 + √3] + (1/4) * sqrt(3) + 1
= 0 - 0 + (1/4) * sqrt(3) + 1
= 1 + (1/4) * sqrt(3)
Поэтому cos 170 sin35 - cos35 sin 170 sin105 sin75 + sin 15 cos 105 = 1 + (1/4) * sqrt(3)
Для начала разберемся с умножением тригонометрических функций:
cos(α)sin(β) = (1/2) * [sin(α + β) - sin(α - β)]
sin(α)sin(β) = (1/2) * [cos(α - β) - cos(α + β)]
Теперь применим эти формулы:
cos 170 sin35 - cos35 sin 170 sin105 sin75 + sin 15 cos 105
= (1/2) [sin(170 + 35) - sin(170 - 35)] - (1/2) [cos(35 - 170) - cos(35 + 170)] * sin105 sin75 + sin 15 cos 105
= (1/2) [sin 205 - sin 135] - (1/2) [-cos 135 - cos 205] (1/2) [cos(105 - 75) - cos(105 + 75)] + (1/2) * [sin(15 + 105) + sin(105 - 15)]
= (1/2) [sin 205 - sin 135] - (1/2) [cos 135 + cos 205] (1/2) [cos 30 - cos 180] + (1/2) * [sin 120 + sin 90]
= (1/2) [sin 205 - sin 135] - (1/2) [cos 135 + cos 205] (1/2) [sqrt(3)/2 - (-1/2)] + (1/2) * [sqrt(3)/2 + 1]
= (1/2) [sin 205 - sin 135] - (1/4) [sqrt(3)(cos 135 + cos 205) + cos 135 + cos 205] + (1/4) * sqrt(3) + 1
= (1/2) [sin 205 - sin 135] - (1/4) [sqrt(3)(-√2/2 + -√2/2) + -√2/2 + √2/2] + (1/4) * sqrt(3) + 1
= (1/2) [-√2/2 - -√2/2] - (1/4) [-√3 + √3] + (1/4) * sqrt(3) + 1
= 0 - 0 + (1/4) * sqrt(3) + 1
= 1 + (1/4) * sqrt(3)
Поэтому cos 170 sin35 - cos35 sin 170 sin105 sin75 + sin 15 cos 105 = 1 + (1/4) * sqrt(3)