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15. Втреутольнике ABC угол \( \mathrm{C}=90^{\circ}, \mathrm{AC}=9, \cos \mathrm{~A}=0,6 \). найди длину стороны AB .

Ask by Rodriguez Powers. in Russia
Mar 14,2025

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Длина стороны \( AB \) равна \(15\).

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Чтобы найти длину стороны \( AB \) в прямоугольном треугольнике \( ABC \) с углом \( C=90^{\circ} \), используем известные значения. Из условия \( \cos A = 0.6 \) находим синус угла \( A \): \[ \sin^2 A + \cos^2 A = 1 \implies \sin^2 A = 1 - \cos^2 A = 1 - 0.6^2 = 1 - 0.36 = 0.64 \implies \sin A = 0.8. \] Теперь, используя соотношения в прямоугольном треугольнике, мы можем выразить стороны: \[ \sin A = \frac{BC}{AC} \implies BC = AC \cdot \sin A = 9 \cdot 0.8 = 7.2. \] Также используя \( \cos A = \frac{AB}{AC} \): \[ AB = AC \cdot \cos A = 9 \cdot 0.6 = 5.4. \] Итак, длина стороны \( AB \) равна \( 5.4 \).

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