Упр.603 ГДЗ Алимов 10-11 класс (Алгебра)

1. $$\sin\left(\arcsin\frac{1}{3}+\arccos\frac{2\sqrt{2}}{3}\right)$$
2. $$\cos\left(\arcsin\frac{3}{5}+\arccos\frac{4}{5}\right)$$

1) Пусть $$\alpha=\arcsin \frac{1}{3}, \quad \beta=\arccos \frac{2\sqrt{2}}{3}.$$ Тогда

$$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta.$$

$$\sin\alpha=\frac{1}{3}, \quad \cos\beta=\frac{2\sqrt{2}}{3},$$
$$\cos\alpha=\sqrt{1-\sin^2\alpha}=\sqrt{1-\frac{1}{9}}=\frac{2\sqrt{2}}{3},$$
$$\sin\beta=\sqrt{1-\cos^2\beta}=\sqrt{1-\frac{8}{9}}=\frac{1}{3}.$$

$$\sin\left(\arcsin \frac{1}{3}+\arccos \frac{2\sqrt{2}}{3}\right)=\frac{1}{3}\cdot \frac{2\sqrt{2}}{3}+\frac{2\sqrt{2}}{3}\cdot \frac{1}{3}=\frac{4\sqrt{2}}{9}.$$

Ответ: $$\frac{4\sqrt{2}}{9}.$$

2) Пусть $$\alpha=\arcsin \frac{3}{5}, \quad \beta=\arccos \frac{4}{5}.$$ Тогда

$$\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta.$$

$$\sin\alpha=\frac{3}{5}, \quad \cos\beta=\frac{4}{5},$$
$$\cos\alpha=\sqrt{1-\sin^2\alpha}=\sqrt{1-\frac{9}{25}}=\frac{4}{5},$$
$$\sin\beta=\sqrt{1-\cos^2\beta}=\sqrt{1-\frac{16}{25}}=\frac{3}{5}.$$

$$\cos\left(\arcsin \frac{3}{5}+\arccos \frac{4}{5}\right)=\frac{4}{5}\cdot \frac{4}{5}-\frac{3}{5}\cdot \frac{3}{5}=\frac{16}{25}-\frac{9}{25}=\frac{7}{25}.$$

Ответ: $$\frac{7}{25}.$$