10. Вычислите sin 2α, cos 2β, sin(α - β) и cos(α + β), если: a) sin α = 4/5, π/2 < α < π, cos β = -5/13, π/2 < β < π б) cos α = 0,6, 3π/2 < α < 2π, sin β = -8/17, π < β < 3π/2

a) sin α = 4/5, cos β = -5/13, π/2 < α < π, π/2 < β < π cos α = -√(1 - sin² α) = -√(1 - 16/25) = -√(9/25) = -3/5 sin β = √(1 - cos² β) = √(1 - 25/169) = √(144/169) = 12/13 sin 2α = 2 sin α cos α = 2 * (4/5) * (-3/5) = -24/25 cos 2β = cos² β - sin² β = 25/169 - 144/169 = -119/169 sin(α - β) = sin α cos β - cos α sin β = (4/5) * (-5/13) - (-3/5) * (12/13) = -20/65 + 36/65 = 16/65 cos(α + β) = cos α cos β - sin α sin β = (-3/5) * (-5/13) - (4/5) * (12/13) = 15/65 - 48/65 = -33/65 б) cos α = 0,6 = 3/5, sin β = -8/17, 3π/2 < α < 2π, π < β < 3π/2 sin α = -√(1 - cos² α) = -√(1 - 9/25) = -√(16/25) = -4/5 cos β = -√(1 - sin² β) = -√(1 - 64/289) = -√(225/289) = -15/17 sin 2α = 2 sin α cos α = 2 * (-4/5) * (3/5) = -24/25 cos 2β = cos² β - sin² β = 225/289 - 64/289 = 161/289 sin(α - β) = sin α cos β - cos α sin β = (-4/5) * (-15/17) - (3/5) * (-8/17) = 60/85 + 24/85 = 84/85 cos(α + β) = cos α cos β - sin α sin β = (3/5) * (-15/17) - (-4/5) * (-8/17) = -45/85 - 32/85 = -77/85