Giả sử A \( = \cos \frac{\pi }{{15}} \cdot \cos \frac{{2\pi }}{{15}} \cdot \cos \frac{{3\pi }}{{15}} \cdot \cos \frac{{4\pi }}{{15}} \cdot \cos \frac{{5\pi }}{{15}} \cdot \cos \frac{{6\pi }}{

21/25

Giả sử A \( = \cos \frac{\pi }{{15}} \cdot \cos \frac{{2\pi }}{{15}} \cdot \cos \frac{{3\pi }}{{15}} \cdot \cos \frac{{4\pi }}{{15}} \cdot \cos \frac{{5\pi }}{{15}} \cdot \cos \frac{{6\pi }}{{15}} \cdot \cos \frac{{7\pi }}{{15}} = \frac{a}{b}\) . Khi đó giá trị của biểu thức a + b bằng bao nhiêu?

Nhập câu trả lời ở đây

Giải thích

Đáp án: 129

\(A = \cos \frac{\pi }{{15}} \cdot \cos \frac{{2\pi }}{{15}} \cdot \cos \frac{{3\pi }}{{15}} \cdot \cos \frac{{4\pi }}{{15}} \cdot \cos \frac{{5\pi }}{{15}} \cdot \cos \frac{{6\pi }}{{15}} \cdot \cos \frac{{7\pi }}{{15}}\)

\( = \frac{{\sin \frac{\pi }{{15}} \cdot \cos \frac{\pi }{{15}} \cdot \cos \frac{{2\pi }}{{15}} \cdot \cos \frac{{3\pi }}{{15}} \cdot \cos \frac{{4\pi }}{{15}} \cdot \cos \frac{{5\pi }}{{15}} \cdot \cos \frac{{6\pi }}{{15}} \cdot \cos \frac{{7\pi }}{{15}} \cdot \sin \frac{{3\pi }}{{15}}}}{{\sin \frac{\pi }{{15}} \cdot \sin \frac{{3\pi }}{{15}}}}\)

\(\)\( = \frac{{\sin \frac{{2\pi }}{{15}} \cdot \cos \frac{{2\pi }}{{15}} \cdot \cos \frac{{4\pi }}{{15}} \cdot \frac{1}{2} \cdot \sin \frac{{6\pi }}{{15}} \cdot \cos \frac{{6\pi }}{{15}} \cdot \cos \frac{{7\pi }}{{15}}}}{{4\sin \frac{\pi }{{15}} \cdot \sin \frac{{3\pi }}{{15}}}}\)

\( = \frac{{\sin \frac{{4\pi }}{{15}} \cdot \cos \frac{{4\pi }}{{15}} \cdot \sin \frac{{12\pi }}{{15}} \cdot \cos \frac{{7\pi }}{{15}}}}{{32\sin \frac{\pi }{{15}} \cdot \sin \frac{{3\pi }}{{15}}}} = = \frac{{ - \sin \frac{{8\pi }}{{15}} \cdot \cos \frac{{8\pi }}{{15}} \cdot \sin \frac{{12\pi }}{{15}}}}{{64\sin \frac{\pi }{{15}} \cdot \sin \frac{{3\pi }}{{15}}}}\)

\( = \frac{{ - \sin \frac{{16\pi }}{{15}} \cdot \sin \frac{{12\pi }}{{15}}}}{{128\sin \frac{\pi }{{15}} \cdot \sin \frac{{3\pi }}{{15}}}} = \frac{1}{{128}}\).

Vậy a + b = 129.