Cho pi/2 < α < pi , sin α = 2/7 . Tính P = sin α + cos α

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Cho \(\frac{\pi }{2} < \alpha < \pi \), \(\sin \alpha = \frac{2}{7}\). Tính \(P = \sin \alpha + \cos \alpha \)

\(P = \frac{{2 - 2\sqrt {11} }}{7}.\)

\(P = \frac{{2 - 3\sqrt 5 }}{7}\).

\(P = \frac{{2 + 3\sqrt 5 }}{7}.\)

\(P = \frac{{2 + 2\sqrt {11} }}{7}.\)

Giải thích

Chọn B

\({\cos ^2}\alpha = 1 - {\sin ^2}\alpha \)\( = 1 - \frac{4}{{49}} = \frac{{45}}{{49}}\)\( \Leftrightarrow \cos \alpha = \pm \frac{{3\sqrt 5 }}{7}\)

Do \(\frac{\pi }{2} < \alpha < \pi \)\( \Rightarrow \cos \alpha = - \frac{{3\sqrt 5 }}{7}\)

\(P = \sin \alpha + \cos \alpha \)\( = \frac{2}{7} + \left( { - \frac{{3\sqrt 5 }}{7}} \right) = \frac{{2 - 3\sqrt 5 }}{7}\)