Odpowiedź:
sinα · tgβ = 12/13
Szczegółowe wyjaśnienie:
a = x
b = 2x + 2
c = 3x - 2
Z twierdzenia Pitagorasa:
a² + b² = c²
x² + (2x + 2)² = (3x - 2)²
x² + 4x² + 8x + 4 = 9x² - 12x + 4
5x² - 9x² + 8x + 12x + 4 - 4 = 0
-4x² + 20x = 0 /:(-4)
x² - 5x = 0
x ∈ N⁺
x(x - 5) = 0
x = 0 ∉ N+ ∨ x - 5 = 0
x = 5
a = 5
b = 2x + 2 = 2·5 + 2 = 10 + 2 = 12
c = 3x - 2 = 3·5 - 2 = 15 - 2 = 13
[tex]sin\alpha = \frac{a}{c} = \frac{5}{13}\\\\tg\beta = \frac{b}{a} = \frac{12}{5}\\\\sin\alpha\cdot tg\beta = \frac{5}{13}\cdot\frac{12}{5} = \frac{12}{13}[/tex]
