Упр.23.1 ГДЗ Мерзляк 10 класс Базовый уровень (Алгебра)

1) cos(a); 3) tg(a/2); 5) sin(2x-?/6); 7) sin(a-?);
2) sin(3a); 4) cos(8a); 6) tg(7a); 8) tg(3).

1. $$\cos a=\cos\left(2\cdot \frac a2\right)=\cos^2\frac a2-\sin^2\frac a2.$$
2. $$\sin 3a=\sin\left(2\cdot \frac{3a}{2}\right)=2\sin\frac{3a}{2}\cdot \cos\frac{3a}{2}.$$
3. $$\tg\frac a2=\tg\left(2\cdot \frac a4\right)=\frac{2\tg\frac a4}{1-\tg^2\frac a4}.$$
4. $$\cos 8a=\cos(2\cdot 4a)=\cos^2 4a-\sin^2 4a.$$
5. $$\sin\left(2x-\frac{\pi}{6}\right)=\sin\left(2\left(x-\frac{\pi}{12}\right)\right)=2\sin\left(x-\frac{\pi}{12}\right)\cos\left(x-\frac{\pi}{12}\right).$$
6. $$\tg 7a=\tg\left(2\cdot \frac{7a}{2}\right)=\frac{2\tg\frac{7a}{2}}{1-\tg^2\frac{7a}{2}}.$$
7. $$\sin(a-\beta)=\sin\left(2\left(\frac a2-\frac \beta2\right)\right)=2\sin\left(\frac a2-\frac \beta2\right)\cos\left(\frac a2-\frac \beta2\right).$$
8. $$\tg 3=\tg(2\cdot 1{,}5)=\frac{2\tg 1{,}5}{1-\tg^2 1{,}5}.$$

Ответ

$$\cos a=\cos^2\frac a2-\sin^2\frac a2;$$
$$\sin 3a=2\sin\frac{3a}{2}\cos\frac{3a}{2};$$
$$\tg\frac a2=\frac{2\tg\frac a4}{1-\tg^2\frac a4};$$
$$\cos 8a=\cos^2 4a-\sin^2 4a;$$
$$\sin\left(2x-\frac{\pi}{6}\right)=2\sin\left(x-\frac{\pi}{12}\right)\cos\left(x-\frac{\pi}{12}\right);$$
$$\tg 7a=\frac{2\tg\frac{7a}{2}}{1-\tg^2\frac{7a}{2}};$$
$$\sin(a-\beta)=2\sin\left(\frac a2-\frac \beta2\right)\cos\left(\frac a2-\frac \beta2\right);$$
$$\tg 3=\frac{2\tg 1{,}5}{1-\tg^2 1{,}5}.$$