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Công thức biến đổi tổng thành tích - Toán 11

1. Công thức biến đổi tổng thành tích

\(\cos a + \cos b = 2\cos \frac{{a + b}}{2}\cos \frac{{a - b}}{2}\); \(\cos a - \cos b =  - 2\sin \frac{{a + b}}{2}\sin \frac{{a - b}}{2}\); \(\sin a + \sin b = 2\sin \frac{{a + b}}{2}\cos \frac{{a - b}}{2}\); \(\sin a - \sin b = 2\cos \frac{{a + b}}{2}\sin \frac{{a - b}}{2}\).

2. Áp dụng dụng công thức biến đổi tổng thành tích vào bài tập

1) Không dùng máy tính, tính giá trị của biểu thức \(A = \sin \frac{\pi }{9} - \sin \frac{{4\pi }}{9} + \sin \frac{{7\pi }}{9}\).

Giải:

\(A = \left( {\sin \frac{\pi }{9} + \sin \frac{{7\pi }}{9}} \right) - \sin \frac{{4\pi }}{9}\) \( = 2\sin \frac{{4\pi }}{9}\cos \frac{\pi }{3} - \sin \frac{{4\pi }}{9} = 2\sin \frac{{4\pi }}{9}.\frac{1}{2} - \sin \frac{{4\pi }}{9} = 0\).

2) Tính \(\sin \frac{{5\pi }}{{12}} + \sin \frac{\pi }{{12}}\).

Giải:

\(\sin \frac{{5\pi }}{{12}} + \sin \frac{\pi }{{12}} = 2\sin \frac{{\frac{{5\pi }}{{12}} + \frac{\pi }{{12}}}}{2}\cos \frac{{\frac{{5\pi }}{{12}} - \frac{\pi }{{12}}}}{2}\) \( = 2\sin \frac{\pi }{4}\cos \frac{\pi }{6} = 2.\frac{{\sqrt 2 }}{2}.\frac{{\sqrt 3 }}{2} = \frac{{\sqrt 6 }}{2}\).

3. Bài tập vận dụng

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