sinα + sin β = √6 /2 oraz α + β = 90⁰ β = 90⁰ - α sin β = sin ( 90⁰ - α) = cos α zatem sin α + sin β = sin α + cos α = √6 /2 ----> cos α = √6 /2 - sin α oraz sin²α + cos²α = 1 sin²α + [ √6 /2 - sin α]² = 1 sin²α + 6/4 - √6 sin α + sin²α = 1 2 sin²α - √6 sin α + 1/2 = 0 Δ = (-√6)² - 4*2*(1/2) = 6 - 4 = 2 √Δ = √2 sin α = [ √6 -√2]/4 lub sin α = [ √6 + √2] /4 Obliczymy teraz kosinus kąta o mierze α cos²α = 1 - sin²α = 1 - [√6 -√2]² / 4² = 1 - [6-2√12 +2]/16 = = [ 16 - 8 + 2√12]/16 = [8 +2√12]/16 = [√6 + √2]² / 4² stąd cos α = [√6 + √2]/ 4 ale cos α = sin β , zatem sin α *sin β = [√6 - √2]/4 * [√6 + √2] / 4 = [ (√6)² -(√2)²]/4² = = [6 - 2]/16 = 4/16 = 1/4 Podobnie można wykazać dla kąta , którego sin α = [√6 +√2]/4. Wtedy cos α = [√6 - √2] /4 i iloczyn sin α * sin β = 1/4 --------------------------------- α = 15⁰ lub α = 75⁰. Odp. sin α* sin β = 1/4
