5. Найти sina, tga, sin2a, cos2x, если cosa = -20/29 , π/2 < a < π

Дано:

• $$ \cos \alpha = -\frac{20}{29} $$
• $$ \frac{\pi}{2} < \alpha < \pi $$

Найти: $$ \sin \alpha, \tan \alpha, \sin 2\alpha, \cos 2\alpha $$

Решение:

1. Так как $$ \frac{\pi}{2} < \alpha < \pi $$, то $$ \sin \alpha > 0 $$.

   Основное тригонометрическое тождество: $$ \sin^2 \alpha + \cos^2 \alpha = 1 $$.

   Выразим $$ \sin \alpha $$:

   $$ \sin \alpha = \sqrt{1 - \cos^2 \alpha} = \sqrt{1 - \left(-\frac{20}{29}\right)^2} = \sqrt{1 - \frac{400}{841}} = \sqrt{\frac{841 - 400}{841}} = \sqrt{\frac{441}{841}} = \frac{21}{29} $$

   Итак, $$ \sin \alpha = \frac{21}{29} $$.

2. Найдем $$ \tan \alpha $$.

   $$ \tan \alpha = \frac{\sin \alpha}{\cos \alpha} = \frac{\frac{21}{29}}{-\frac{20}{29}} = -\frac{21}{20} $$

   Итак, $$ \tan \alpha = -\frac{21}{20} $$.

3. Найдем $$ \sin 2\alpha $$.

   $$ \sin 2\alpha = 2 \sin \alpha \cos \alpha = 2 \cdot \frac{21}{29} \cdot \left(-\frac{20}{29}\right) = -\frac{2 \cdot 21 \cdot 20}{29^2} = -\frac{840}{841} $$

   Итак, $$ \sin 2\alpha = -\frac{840}{841} $$.

4. Найдем $$ \cos 2\alpha $$.

   $$ \cos 2\alpha = \cos^2 \alpha - \sin^2 \alpha = \left(-\frac{20}{29}\right)^2 - \left(\frac{21}{29}\right)^2 = \frac{400}{841} - \frac{441}{841} = -\frac{41}{841} $$

   Итак, $$ \cos 2\alpha = -\frac{41}{841} $$.

Ответ: $$ \sin \alpha = \frac{21}{29}, \tan \alpha = -\frac{21}{20}, \sin 2\alpha = -\frac{840}{841}, \cos 2\alpha = -\frac{41}{841} $$