Question

Cho \(\cos 2a = \frac{1}{3}\) với \(\frac{\pi }{2} < a < \pi \). Tính \(\sin a,\,\,\cos a,\,\,\tan a\)

Answer 1

\(\begin{array}{l}\cos 2a = \frac{1}{3} \Leftrightarrow {\cos ^2}a - {\sin ^2}a = \frac{1}{3}\,\,\left( 1 \right)\\{\cos ^2}a + {\sin ^2}a = 1\,\,\,\,\left( 2 \right)\end{array}\)

Từ (1) và (2) \( \Rightarrow \left\{ \begin{array}{l}{\cos ^2}a = \frac{2}{3}\\{\sin ^2}a = \frac{1}{3}\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}\cos a =  \pm \frac{{\sqrt 6 }}{3}\\\sin a =  \pm \frac{{\sqrt 3 }}{3}\end{array} \right.\)

Do \(\frac{\pi }{2} < a < \pi \)\( \Rightarrow \left\{ \begin{array}{l}\cos a = \frac{{-\sqrt 6 }}{3}\\\sin a =  \ \frac{{\sqrt 3 }}{3}\end{array} \right.\)

\(\Rightarrow \tan a = \frac{{\sin a}}{{\cos a}} =  - \frac{{\sqrt 2 }}{2}\)