587. Найдите \(\sin \alpha\), \(\tan \alpha\) и \(\cot \alpha\), если \(\cos \alpha = \frac{1}{3}\).

Решение: 1. Найдем \(\sin \alpha\) используя основное тригонометрическое тождество: \(\sin^2 \alpha + \cos^2 \alpha = 1\) \(\sin^2 \alpha = 1 - \cos^2 \alpha = 1 - (\frac{1}{3})^2 = 1 - \frac{1}{9} = \frac{8}{9}\) \(\sin \alpha = \sqrt{\frac{8}{9}} = \frac{2\sqrt{2}}{3}\) 2. Найдем \(\tan \alpha\): \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha} = \frac{\frac{2\sqrt{2}}{3}}{\frac{1}{3}} = 2\sqrt{2}\) 3. Найдем \(\cot \alpha\): \(\cot \alpha = \frac{1}{\tan \alpha} = \frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4}\) Ответ: \(\sin \alpha = \frac{2\sqrt{2}}{3}\), \(\tan \alpha = 2\sqrt{2}\), \(\cot \alpha = \frac{\sqrt{2}}{4}\)