Упр.545 ГДЗ Алимов 10-11 класс (Алгебра)

Разложить на множители:

1. $$1-\cos a+\sin a$$;
2. $$1-2\cos a+\cos 2a$$;
3. $$1+\sin a-\cos a-\tan a$$;
4. $$1+\sin a+\cos a+\tan a$$.

1) $$1-\cos \alpha+\sin \alpha=(\cos^2 \frac{\alpha}{2}+\sin^2 \frac{\alpha}{2})-(\cos^2 \frac{\alpha}{2}-\sin^2 \frac{\alpha}{2})+2\sin \frac{\alpha}{2}\cos \frac{\alpha}{2}$$

$$=2\sin^2 \frac{\alpha}{2}+2\sin \frac{\alpha}{2}\cos \frac{\alpha}{2}=2\sin \frac{\alpha}{2}\left(\sin \frac{\alpha}{2}+\cos \frac{\alpha}{2}\right).$$

2) $$1-2\cos \alpha+\cos 2\alpha=(\cos^2 \alpha+\sin^2 \alpha)+(\cos^2 \alpha-\sin^2 \alpha)-2\cos \alpha$$

$$=2\cos^2 \alpha-2\cos \alpha=2\cos \alpha(\cos \alpha-1).$$

3) $$1+\sin \alpha-\cos \alpha-\tg \alpha=\frac{\cos \alpha+\sin \alpha\cos \alpha-\cos^2 \alpha-\sin \alpha}{\cos \alpha}$$

$$=\frac{\cos \alpha(1-\cos \alpha)-\sin \alpha(1-\cos \alpha)}{\cos \alpha}$$

$$=\frac{(1-\cos \alpha)(\cos \alpha-\sin \alpha)}{\cos \alpha}=(1-\cos \alpha)(1-\tg \alpha).$$

4) $$1+\sin \alpha+\cos \alpha+\tg \alpha=\frac{\cos \alpha+\sin \alpha\cos \alpha+\cos^2 \alpha+\sin \alpha}{\cos \alpha}$$

$$=\frac{\cos \alpha(1+\cos \alpha)+\sin \alpha(1+\cos \alpha)}{\cos \alpha}$$

$$=\frac{(\cos \alpha+\sin \alpha)(1+\cos \alpha)}{\cos \alpha}=(1+\tg \alpha)(1+\cos \alpha).$$

Ответ

1) $$2\sin \frac{\alpha}{2}\left(\sin \frac{\alpha}{2}+\cos \frac{\alpha}{2}\right);$$ 2) $$2\cos \alpha(\cos \alpha-1);$$ 3) $$(1-\cos \alpha)(1-\tg \alpha);$$ 4) $$(1+\tg \alpha)(1+\cos \alpha).$$