Тригонометрия. Урок 2. Группировка и разложение на множители.

Домашнее задание из В.В. Ткачук “Математика – абитуриенту”

1. $$\sin x\cdot(3\sin 2x\sin^3 x+12\sin 2x\sin x-16\cos x)+2\sin 4x=0$$
2. $$3\cos 4x+2\cos 2x\cdot (10\cos^4 x+3\cos^2 x+\sin^2 x)+3=0$$
3. $$(1+2\sin x)\sin x=\sin 2x+\cos x$$
4. $$\sin x+\sin 2x+(1+2\cos x)\cos x=0$$
5. [3] $$6tg x+5ctg 3x=tg 2x$$
6. $$\sin 3x\cos x=\frac{3}{2}tg x$$
7. $$\cos x+\sin x=\frac{\cos 2x}{1-\sin 2x}$$
8. [3] $$\frac{1}{\sin x}-\frac{1}{\cos x}=2\sin\frac{\pi}{4}$$
9. $$2(\sin^3 x+\cos^3 x)=3\sin 2x\cdot (\sin x+\cos x)$$
10. $$\sin 2x\cdot (tg^2 x+3)=4(\cos 2x-1)$$
11. $$16\sin x-\sin 2x=1-\cos 2x$$
12. [3] $$4\sin x+2\cos x=2+3tg x$$
13. [2] $$1+\sin^3 x+\cos^3 x=\frac{3}{2}\sin 2x$$
14. $$3tg^2 x+4tg x+4ctg x+3ctg^2 x+2=0$$
15. $$\sqrt{3}(\cos 2x +1)=2\cos x\cdot (2-\cos 2x)$$
16. $$2(\cos x-1)\sin 2x=3\sin x$$
17. [3] $$3\sin x\cos x+4\sin x=4-3\cos^2 x+\cos x$$
18. $$2\sin x\cdot tg x+2tg x=\sin x+3\cos x+3$$
19. [2] $$2\sin x-4\cos x-tg x=ctg x=2\cos x\cdot ctg x-2$$
20. $$1-5\sin x=5ctg x-\frac{\sin^2 x}{\cos x}$$

Ответы к домашнему заданию урока 2 из В.В. Ткачук “Математика – абитуриенту”

   По умолчанию, $$n \in Z$$.

1. $$\pi n/2, \pm (-1)^n arcsin \sqrt{2/3}+n\pi$$
2. $$\pi/4+n\pi/2, \pm arccos(\pm \frac{1}{\sqrt{5}})+2n\pi$$
3. $$\pi/4+n\pi, (-1)^{n+1}\pi/6+n\pi$$
4. $$-\pi/4+n\pi, \pm 2\pi/3+2n\pi$$
5. $$\pm arccos(1/3)/2+n\pi, \pm arccos(-1/4)/2+n\pi$$
6. $$n\pi, \pm \pi/6+n\pi$$
7. $$-\pi/4+n\pi, -\pi/4\pm \pi/4+2n\pi$$
8. $$-\pi/4 \pm arccos((\sqrt{3}-1)/2)+2n\pi$$
9. $$-\pi/4+n\pi, (-1)^n\pi/12+n\pi/2$$
10. $$n\pi, -\pi/4+n\pi, -arctg3+n\pi$$
11. $$n\pi$$
12. $$(-1)^n\pi/6+n\pi, 2n\pi, -2arctg(1/2)+2n\pi$$
13. $$-\pi/4+(-1)^{n+1}\pi/4+n\pi$$
14. $$-\pi/4+n\pi$$
15. $$\pi/2+n\pi, \pm \pi/6+2n\pi$$
16. $$n\pi, \pm 2\pi/3+2n\pi$$
17. $$(-1)^n arcsin(1/3)+n\pi, -\pi/4+(\pm 3\pi/4+2n\pi)$$
18. $$\pi+2n\pi, arctg(3/2)+n\pi$$
19. $$\pm\pi/3+2n\pi, \pi/4+n\pi$$
20. $$\pm arccos((1-\sqrt{5})/2)+2n\pi, arctg5+n\pi$$