Тригонометрические формулы

$$tg\alpha=\frac{\sin\alpha}{\cos\alpha},\;\;ctg\alpha=\frac{\cos\alpha}{\sin\alpha},$$ $$\sin(-x)=-\sin(x),\;\;\cos(-x)=\cos(x),$$ $$tg(-x)=-tg(x),\;\;ctg(-x)=-ctg(x),$$ $$\sin^2\alpha+\cos^2\alpha=1,$$ $$\sin^2\alpha=1-\cos^2\alpha,$$ $$\cos^2\alpha=1-\sin^2\alpha,$$ $$1+tg^2\alpha=\frac{1}{\cos^2\alpha},$$ $$1+ctg^2\alpha=\frac{1}{\sin^2\alpha},$$ $$\sin\alpha=\pm\sqrt{1-\cos^2\alpha},$$ $$\cos\alpha=\pm\sqrt{1-\sin^2\alpha},$$
$$\cos(\alpha+\beta)=\cos\alpha\cdot\cos\beta-\sin\alpha\cdot\sin\beta,$$ $$\cos(\alpha-\beta)=\cos\alpha\cdot\cos\beta+\sin\alpha\cdot\sin\beta,$$ $$\sin(\alpha+\beta)=\sin\alpha\cdot\cos\beta+\sin\beta\cdot\cos\alpha,$$ $$\sin(\alpha-\beta)=\sin\alpha\cdot\cos\beta-\sin\beta\cdot\cos\alpha,$$ $$tg(\alpha+\beta)=\frac{tg\alpha+tg\beta}{1-tg\alpha\cdot tg\beta},$$ $$tg(\alpha-\beta)=\frac{tg\alpha-tg\beta}{1+tg\alpha\cdot tg\beta},$$ $$ctg(\alpha+\beta)=\frac{ctg\alpha\cdot ctg\beta-1}{ctg\alpha+ ctg\beta},$$ $$ctg(\alpha-\beta)=\frac{ctg\alpha\cdot ctg\beta+1}{ctg\beta- ctg\alpha},$$ $$\sin2\alpha=2\sin\alpha\cos\alpha;$$ $$\cos2\alpha=\cos^2\alpha-\sin^2\alpha=2\cos^2\alpha-1=1-2\sin^2\alpha;$$ $$1+\cos2\alpha=2\cos^2\alpha;$$ $$1-\cos2\alpha=2\sin^2\alpha;$$ $$tg2\alpha=\frac{2tg\alpha}{1-tg^2\alpha};$$ $$ctg2\alpha=\frac{ctg^2\alpha-1}{2ctg\alpha};$$ $$\cos^2\alpha=\frac{1}{2}(1+\cos2\alpha);$$ $$\sin^2\alpha=\frac{1}{2}(1-\cos2\alpha);$$ $$\sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2};$$ $$\sin\alpha-\sin\beta=2\sin\frac{\alpha-\beta}{2}\cos\frac{\alpha+\beta}{2};$$ $$\cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2};$$ $$\cos\alpha-\cos\beta=-2\sin\frac{\alpha-\beta}{2}\sin\frac{\alpha+\beta}{2};$$ $$\sin\alpha\cdot\sin\beta=\frac{1}{2}(\cos(\alpha-\beta)-\cos(\alpha+\beta));$$ $$\cos\alpha\cdot\cos\beta=\frac{1}{2}(\cos(\alpha-\beta)+\cos(\alpha+\beta));$$ $$\sin\alpha\cdot\cos\beta=\frac{1}{2}(\sin(\alpha+\beta)+\sin(\alpha-\beta));$$ $$tg\frac{\alpha}{2}=\frac{\sin\alpha}{1+\cos\alpha}=\frac{1-\cos\alpha}{\sin\alpha};$$ $$ctg\frac{\alpha}{2}=\frac{1+\cos\alpha}{\sin\alpha}=\frac{\sin\alpha}{1-\cos\alpha};$$ $$\sin\alpha=\frac{2tg\frac{\alpha}{2}}{1+tg^2\frac{\alpha}{2}};$$ $$\cos\alpha=\frac{1-tg^2\frac{\alpha}{2}}{1+tg^2\frac{\alpha}{2}}.$$ $$\sin(\arcsin x)=x,\;\;\cos(\arccos x)=x,\;\; x\in [-1; 1];$$ $$tg(arctg x)=x,\;\; ctg(arcctg x)=x,\;\; x\in R;$$ $$\arcsin(\sin x)=x,\;\; x\in[-\frac{\pi}{2}; \frac{\pi}{2}];\;\; \arccos(\cos x)=x,\;\; x\in[0; \pi];$$ $$arctg(tg x)=x,\;\; x\in (-\frac{\pi}{2}; \frac{\pi}{2});\;\; arcctg(ctg x)=x,\;\; x\in(0; \pi);$$ $$\arcsin x+\arccos x=\frac{\pi}{2},\;\; arctg x+arcctg x=\frac{\pi}{2};$$ $$\arcsin(-x)=-\arcsin x,\;\; \arccos(-x)=\pi-\arccos x;$$ $$arctg(-x)=-arctg x,\;\; arcctg(-x)=\pi-arcctg x;$$ $$1) \sin x=A;$$ $$Если\;\; |A|>1, \;\; x\in\emptyset;$$ $$Если\;\; |A|\leq 1,\;\; x=(-1)^n\cdot\arcsin A+\pi n,\;n\in Z;$$ $$2) \cos x=A;$$ $$Если\;\; |A|>1, \;\; x\in\emptyset;$$ $$Если\;\; |A|\leq 1,\;\; x=\pm\arccos A+2\pi n,\;n\in Z;$$ $$3) tg x=A;\;\; x=arctg A+\pi n, \;n\in Z;$$ $$4) ctg x=A;\;\; x=arcctg A+\pi n,\; n\in Z;$$