1. W trójkącie prostokątnym a, b oznaczają długości przyprostokątnych, c jest długością przeciwprostokątnej, α oznacza miarę kąta leżącego naprzeciw przyprostokątnej a. Wiedząc że sinα = √5/3 oblicz:
b)wartość wyrażenia a²/ 2b²+c²
2. Kąt α jest kątem ostrym oraz sin α + cos α = 1 1/4. Oblicz:
a) (sinα-cosα)²
b) sin⁴α+cos⁴α
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1.![(\frac{\sqrt{5}}{3})^{2}+cos^{2}\alpha = 1\\cos\alpha = \frac{2}{3} = \frac{b}{c}\\b = \frac{2}{3}c\\b^{2} = \frac{4}{9}c^{2}\\\\\frac{\sqrt{5}}{3}=\frac{a}{c}\\a = \frac{\sqrt{5}}{3}c\\a^{2} = \frac{5}{9}c^{2}\\\\\frac{a^{2}}{2b^{2}+c^{2}} = \frac{\frac{5}{9}c^{2}}{2*\frac{4}{9}c^{2}+c^{2}} = \frac{5}{9}*\frac{9}{17} = \frac{5}{17} (\frac{\sqrt{5}}{3})^{2}+cos^{2}\alpha = 1\\cos\alpha = \frac{2}{3} = \frac{b}{c}\\b = \frac{2}{3}c\\b^{2} = \frac{4}{9}c^{2}\\\\\frac{\sqrt{5}}{3}=\frac{a}{c}\\a = \frac{\sqrt{5}}{3}c\\a^{2} = \frac{5}{9}c^{2}\\\\\frac{a^{2}}{2b^{2}+c^{2}} = \frac{\frac{5}{9}c^{2}}{2*\frac{4}{9}c^{2}+c^{2}} = \frac{5}{9}*\frac{9}{17} = \frac{5}{17}](https://tex.z-dn.net/?f=%28%5Cfrac%7B%5Csqrt%7B5%7D%7D%7B3%7D%29%5E%7B2%7D%2Bcos%5E%7B2%7D%5Calpha+%3D+1%5C%5Ccos%5Calpha+%3D+%5Cfrac%7B2%7D%7B3%7D+%3D+%5Cfrac%7Bb%7D%7Bc%7D%5C%5Cb+%3D+%5Cfrac%7B2%7D%7B3%7Dc%5C%5Cb%5E%7B2%7D+%3D+%5Cfrac%7B4%7D%7B9%7Dc%5E%7B2%7D%5C%5C%5C%5C%5Cfrac%7B%5Csqrt%7B5%7D%7D%7B3%7D%3D%5Cfrac%7Ba%7D%7Bc%7D%5C%5Ca+%3D+%5Cfrac%7B%5Csqrt%7B5%7D%7D%7B3%7Dc%5C%5Ca%5E%7B2%7D+%3D+%5Cfrac%7B5%7D%7B9%7Dc%5E%7B2%7D%5C%5C%5C%5C%5Cfrac%7Ba%5E%7B2%7D%7D%7B2b%5E%7B2%7D%2Bc%5E%7B2%7D%7D+%3D+%5Cfrac%7B%5Cfrac%7B5%7D%7B9%7Dc%5E%7B2%7D%7D%7B2%2A%5Cfrac%7B4%7D%7B9%7Dc%5E%7B2%7D%2Bc%5E%7B2%7D%7D+%3D+%5Cfrac%7B5%7D%7B9%7D%2A%5Cfrac%7B9%7D%7B17%7D+%3D+%5Cfrac%7B5%7D%7B17%7D)
2. a) (sinα-cosα)(sinα-cosα) = sin²α - cosαsinα-cosαsinα-cos²α = sin²α-2cosαsinα+cos²α=1-2cosαsinα
(sinα+cosα)² = (⁵/₄)²
1+2cosαsinα = ²⁵/₁₆
2cosαsinα = ⁹/₁₆
1-2* ⁹/₁₆ = ⁷/₁₆
b) (sinα+cosα)² = (⁵/₄)²
1+2cosαsinα = ²⁵/₁₆
2cosαsinα = ⁹/₁₆
cosαsinα = ⁹/₃₂
(cos²α+sin²α)-2(sinαcosα)² = 1-2*(⁹/₃₂)² = ⁴³¹/₅₁₂