Упр.28.322 ГДЗ Мерзляк 11 класс Углубленный уровень (Алгебра)
- Упростите выражение:
1) $$\sin\left(a+\frac{\pi}{2}\right)\cos(\pi-a)+\cos\left(a+\frac{3\pi}{2}\right)\sin(2\pi-a)$$;
2) $$\frac{\sin(180^\circ-a)\cos(180^\circ+a)\operatorname{tg}(180^\circ-a)}{\sin(270^\circ-a)\operatorname{ctg}(270^\circ+a)\cos(90^\circ+a)}$$;
3) $$\left(\operatorname{tg}\left(\frac{5\pi}{2}+a\right)\sin(2\pi-a)+\sin(3\pi-a)\right)^2-\frac{2\cos^2(\pi-a)}{\operatorname{ctg}(a-\pi)}$$;
4) $$\frac{\sin^2\left(\frac{3\pi}{2}+x\right)}{\operatorname{ctg}^2(x-2\pi)}+\frac{\sin^2(-x)}{\operatorname{ctg}^2\left(x-\frac{3\pi}{2}\right)}$$;
5) $$\frac{\operatorname{tg}\left(\frac{\pi}{2}-a\right)\cos\left(\frac{3\pi}{2}-a\right)\cos(2\pi-a)}{\operatorname{ctg}(\pi+a)\sin\left(\frac{3\pi}{2}+a\right)}$$;
6) $$\frac{\sin(\pi+a)}{\sin\left(\frac{3\pi}{2}-a\right)}-\frac{\operatorname{tg}\left(\frac{3\pi}{2}+a\right)}{\operatorname{ctg}(\pi-a)}+\operatorname{tg}(\pi-a)$$.
$$\sin\left(a+\frac{\pi}{2}\right)\cos(\pi-a)+\cos\left(a+\frac{3\pi}{2}\right)\sin(2\pi-a)$$
$$=\cos a\cdot(-\cos a)+\sin a\cdot(-\sin a)$$
$$=-(\cos^2 a+\sin^2 a)=-1.$$$$\frac{\sin(180^\circ-a)\cos(180^\circ+a)\tg(180^\circ-a)}{\sin(270^\circ-a)\ctg(270^\circ+a)\cos(90^\circ+a)}$$
$$=\frac{\sin a\cdot(-\cos a)\cdot(-\tg a)}{-\cos a\cdot(-\tg a)\cdot(-\sin a)}=-1.$$$$\left(\tg\left(\frac{5\pi}{2}+a\right)\sin(2\pi-a)+\sin(3\pi-a)\right)^2-\frac{2\cos^2(\pi-a)}{\ctg(a-\pi)}$$
$$=\left(-\ctg a\cdot(-\sin a)+\sin a\right)^2-\frac{2\cos^2 a}{\ctg a}$$
$$=\left(\frac{\cos a}{\sin a}\sin a+\sin a\right)^2-2\cos^2 a:\frac{\cos a}{\sin a}$$
$$=(\cos a+\sin a)^2-2\cos a\sin a$$
$$=\cos^2 a+2\cos a\sin a+\sin^2 a-2\cos a\sin a=1.$$$$\frac{\sin^2\left(\frac{3\pi}{2}+x\right)}{\ctg^2(x-2\pi)}+\frac{\sin^2(-x)}{\ctg^2\left(x-\frac{3\pi}{2}\right)}$$
$$=\frac{\cos^2 x}{\ctg^2 x}+\frac{\sin^2 x}{\tg^2 x}$$
$$=\cos^2 x:\frac{\cos^2 x}{\sin^2 x}+\sin^2 x:\frac{\sin^2 x}{\cos^2 x}$$
$$=\sin^2 x+\cos^2 x=1.$$$$\frac{\tg\left(\frac{\pi}{2}-a\right)\cos\left(\frac{3\pi}{2}-a\right)\cos(2\pi-a)}{\ctg(\pi+a)\sin\left(\frac{3\pi}{2}+a\right)}$$
$$=\frac{\ctg a\cdot(-\sin a)\cdot\cos a}{\ctg a\cdot(-\cos a)}=\sin a.$$$$\frac{\sin(\pi+a)}{\sin\left(\frac{3\pi}{2}-a\right)}-\frac{\tg\left(\frac{3\pi}{2}+a\right)}{\ctg(\pi-a)}+\tg(\pi-a)$$
$$=\frac{-\sin a}{-\cos a}-\frac{-\ctg a}{-\ctg a}+(-\tg a)$$
$$=\tg a-1-\tg a=-1.$$
Ответ
1) $$-1$$; 2) $$-1$$; 3) $$1$$; 4) $$1$$; 5) $$\sin a$$; 6) $$-1$$.
