Упр.20.9 ГДЗ Мерзляк 10 класс Базовый уровень (Алгебра)

1) cos a=1/2; 3) tg a=2 и ? < a < 3?/2; 2) sin a=0,6 и ?/2 < a < ?; 4) ctg a=-4/3 и 3?/2 < a < 2?.

1. Если $$\cos a=\frac12,$$ то

   $$\sin a=\pm\sqrt{1-\cos^2 a}=\pm\sqrt{1-\left(\frac12\right)^2}=\pm\frac{\sqrt3}{2}.$$

   Тогда
   $$\tg a=\frac{\sin a}{\cos a}=\frac{\pm \frac{\sqrt3}{2}}{\frac12}=\pm\sqrt3,$$
   $$\ctg a=\frac{1}{\tg a}=\pm\frac{\sqrt3}{3}.$$

2. Если $$\sin a=0{,}6$$ и $$\frac{\pi}{2}

   $$\cos a=-\sqrt{1-\sin^2 a}=-\sqrt{1-0{,}6^2}=-\sqrt{1-0{,}36}=-\sqrt{0{,}64}=-0{,}8.$$

   $$\tg a=\frac{\sin a}{\cos a}=\frac{0{,}6}{-0{,}8}=-0{,}75=-\frac34,$$
   $$\ctg a=\frac{1}{\tg a}=-\frac43.$$

3. Если $$\tg a=2$$ и $$\pi

   $$\cos a=-\sqrt{\frac{1}{1+\tg^2 a}}=-\sqrt{\frac{1}{1+2^2}}=-\sqrt{\frac15}=-\frac{\sqrt5}{5}.$$

   $$\sin a=\tg a\cdot \cos a=2\cdot\left(-\frac{\sqrt5}{5}\right)=-\frac{2\sqrt5}{5},$$
   $$\ctg a=\frac{1}{\tg a}=\frac12.$$

4. Если $$\ctg a=-\frac43$$ и $$\frac{3\pi}{2}0.$$

   $$\sin a=-\sqrt{\frac{1}{1+\ctg^2 a}}=-\sqrt{\frac{1}{1+\left(-\frac43\right)^2}}=-\sqrt{\frac{1}{1+\frac{16}{9}}}=-\sqrt{\frac{9}{25}}=-\frac35.$$

   $$\cos a=\ctg a\cdot \sin a=-\frac43\cdot\left(-\frac35\right)=\frac45,$$
   $$\tg a=\frac{1}{\ctg a}=\frac{1}{-\frac43}=-\frac34.$$

Ответ

1) $$\sin a=\pm\frac{\sqrt3}{2},\ \tg a=\pm\sqrt3,\ \ctg a=\pm\frac{\sqrt3}{3};$$
2) $$\cos a=-0{,}8,\ \tg a=-\frac34,\ \ctg a=-\frac43;$$
3) $$\cos a=-\frac{\sqrt5}{5},\ \sin a=-\frac{2\sqrt5}{5},\ \ctg a=\frac12;$$
4) $$\sin a=-\frac35,\ \cos a=\frac45,\ \tg a=-\frac34.$$