Упр.58 Повторение ГДЗ Колмогоров 10-11 класс (Алгебра)

а) cos^4(?)+sin^4(?), если sin(2?)=2/3;
б) (1-2sin^2(?/2))/(1+sin(?)), если tg(?/2)=m;
в) cos(?), если sin(?)tg(?)=1/2;
г) sin(?), cos(2?), cos(?/2), если tg(?/2)=-v2, ? < ? < 3?/2;

1. $$S=\cos^4\alpha+\sin^4\alpha$$

   $$S=(\cos^2\alpha+\sin^2\alpha)^2-2\sin^2\alpha\cos^2\alpha$$
   $$=1-\frac12\sin^2 2\alpha$$
   $$=1-\frac12\cdot\frac49=\frac79$$

2. $$S=\frac{1-2\sin^2\frac{\alpha}{2}}{1+\sin\alpha}, \quad \tg\frac{\alpha}{2}=m$$

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

   $$S=\frac{\cos\alpha}{1+\sin\alpha}$$

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

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

   $$S=\frac{\cos\frac{\alpha}{2}-\sin\frac{\alpha}{2}}{\cos\frac{\alpha}{2}+\sin\frac{\alpha}{2}}$$
   $$=\frac{1-\tg\frac{\alpha}{2}}{1+\tg\frac{\alpha}{2}}=\frac{1-m}{1+m}$$

3. $$\sin\alpha\cdot\tg\alpha=\frac12$$

   $$\frac{\sin^2\alpha}{\cos\alpha}=\frac12$$
   $$\frac{1-\cos^2\alpha}{\cos\alpha}=\frac12$$
   $$2\cos^2\alpha+\cos\alpha-2=0$$

   $$D=1+16=17$$
   $$\cos\alpha=\frac{-1\pm\sqrt{17}}{4}$$

   Так как $$\sin\alpha\cdot\tg\alpha>0,$$ то $$\cos\alpha>0.$$ Следовательно,
   $$\cos\alpha=\frac{\sqrt{17}-1}{4}.$$

4. $$\tg\frac{\alpha}{2}=-\sqrt2,\quad \pi<\alpha<\frac{3\pi}{2}$$

   $$\cos\frac{\alpha}{2}=-\sqrt{\frac{1}{1+\tg^2\frac{\alpha}{2}}} =-\sqrt{\frac{1}{1+2}}=-\frac{\sqrt3}{3}$$

   $$\sin\frac{\alpha}{2}=\tg\frac{\alpha}{2}\cdot\cos\frac{\alpha}{2} =-\sqrt2\cdot\left(-\frac{\sqrt3}{3}\right)=\frac{\sqrt6}{3}$$

   $$\sin\alpha=2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2} =2\cdot\frac{\sqrt6}{3}\cdot\left(-\frac{\sqrt3}{3}\right) =-\frac{2\sqrt2}{3}$$

   $$\cos 2\alpha=1-2\sin^2\alpha =1-2\cdot\frac{8}{9}=-\frac79$$

   $$\cos\frac{\alpha}{2}=-\frac{\sqrt3}{3}$$

Ответ

а) $$\frac79$$; б) $$\frac{1-m}{1+m}$$; в) $$\frac{\sqrt{17}-1}{4}$$; г) $$\sin\alpha=-\frac{2\sqrt2}{3},\ \cos 2\alpha=-\frac79,\ \cos\frac{\alpha}{2}=-\frac{\sqrt3}{3}.$$