Giá trị của biểu thức A= cos 2pi/ 7 + cos 4pi/ 7 + cos 6pi/ 7 bằng

Hướng dẫn giải:

Đáp án đúng là: B

\(A = cos\frac{{2\pi }}{7} + cos\frac{{4\pi }}{7} + cos\frac{{6\pi }}{7}\)\( = \frac{{\sin \frac{\pi }{7}\left( {cos\frac{{2\pi }}{7} + cos\frac{{4\pi }}{7} + cos\frac{{6\pi }}{7}} \right)}}{{\sin \frac{\pi }{7}}}\)

\[ = \frac{{\sin \frac{\pi }{7}cos\frac{{2\pi }}{7} + \sin \frac{\pi }{7}cos\frac{{4\pi }}{7} + \sin \frac{\pi }{7}cos\frac{{6\pi }}{7}}}{{\sin \frac{\pi }{7}}}\]

Ta có \[\sin \frac{\pi }{7}cos\frac{{2\pi }}{7}\]\[ = \frac{1}{2}\left[ {sin\left( {\frac{\pi }{7} - \frac{{2\pi }}{7}} \right) + \sin \left( {\frac{\pi }{7} + \frac{{2\pi }}{7}} \right)} \right]\]

\[ = \frac{1}{2}\left[ {sin\left( { - \frac{\pi }{7}} \right) + \sin \frac{{3\pi }}{7}} \right]\]\[ = \frac{1}{2}\left( { - sin\frac{\pi }{7} + \sin \frac{{3\pi }}{7}} \right)\].

\[\sin \frac{\pi }{7}cos\frac{{4\pi }}{7}\]\[ = \frac{1}{2}\left[ {\sin \left( {\frac{\pi }{7} - \frac{{4\pi }}{7}} \right) + \sin \left( {\frac{\pi }{7} + \frac{{4\pi }}{7}} \right)} \right]\]

\[ = \frac{1}{2}\left[ {\sin \left( { - \frac{{3\pi }}{7}} \right) + \sin \frac{{5\pi }}{7}} \right]\]\[ = \frac{1}{2}\left( { - \sin \frac{{3\pi }}{7} + \sin \frac{{5\pi }}{7}} \right)\].

\[\sin \frac{\pi }{7}cos\frac{{6\pi }}{7}\]\[ = \frac{1}{2}\left[ {\sin \left( {\frac{\pi }{7} - \frac{{6\pi }}{7}} \right) + \sin \left( {\frac{\pi }{7} + \frac{{6\pi }}{7}} \right)} \right]\]

\[ = \frac{1}{2}\left[ {\sin \left( { - \frac{{5\pi }}{7}} \right) + \sin \pi } \right]\]\[ = \frac{1}{2}\left( { - \sin \frac{{5\pi }}{7} + \sin \pi } \right)\].

Do đó \[\sin \frac{\pi }{7}cos\frac{{2\pi }}{7} + \sin \frac{\pi }{7}cos\frac{{4\pi }}{7} + \sin \frac{\pi }{7}cos\frac{{6\pi }}{7}\]

\[ = \frac{1}{2}\left( { - sin\frac{\pi }{7} + \sin \frac{{3\pi }}{7}} \right)\]\[ + \frac{1}{2}\left( { - \sin \frac{{3\pi }}{7} + \sin \frac{{5\pi }}{7}} \right)\]\[ + \frac{1}{2}\left( { - \sin \frac{{5\pi }}{7} + \sin \pi } \right)\]

\[ = \frac{1}{2}\left( { - sin\frac{\pi }{7} + \sin \pi } \right)\]\[ = - \frac{1}{2}sin\frac{\pi }{7}\].

Do đó \[A = \frac{{ - \frac{1}{2}sin\frac{\pi }{7}}}{{sin\frac{\pi }{7}}} = - \frac{1}{2}\].