sinβ= 5/6. a = 18, b = 30. Найдите c.

По теореме синусов:

$$\frac{b}{sin \beta} = \frac{a}{sin \alpha}$$ $$\frac{30}{\frac{5}{6}} = \frac{18}{sin \alpha}$$ $$30 \cdot \frac{6}{5} = \frac{18}{sin \alpha}$$ $$36 = \frac{18}{sin \alpha}$$ $$sin \alpha = \frac{18}{36}$$ $$sin \alpha = \frac{1}{2}$$ $$\alpha = arcsin \frac{1}{2}$$ $$\alpha = 30^\circ$$

Сумма углов в треугольнике равна 180°:

$$\alpha + \beta + \gamma = 180^\circ$$ $$\gamma = 180^\circ - \alpha - \beta$$ $$\gamma = 180^\circ - 30^\circ - arcsin(\frac{5}{6})$$ $$\gamma = 150^\circ - arcsin(\frac{5}{6})$$

По теореме синусов:

$$\frac{b}{sin \beta} = \frac{c}{sin \gamma}$$ $$c = \frac{b \cdot sin \gamma}{sin \beta}$$ $$c = \frac{30 \cdot sin (150^\circ - arcsin(\frac{5}{6}))}{\frac{5}{6}}$$ $$c = \frac{30 \cdot sin (150^\circ - arcsin(\frac{5}{6})) \cdot 6}{5}$$ $$c = 36 \cdot sin (150^\circ - arcsin(\frac{5}{6}))$$ $$c \approx 9.1$$

Ответ: с ≈ 9.1