Công thức biến đổi tích thành tổng - Toán 11

\(\cos a\cos b = \frac{1}{2}\left[ {\cos (a - b) + \cos (a + b)} \right]\);

\(\sin a\sin b = \frac{1}{2}\left[ {\cos (a - b) - \cos (a + b)} \right]\);

\(\sin a\cos b = \frac{1}{2}\left[ {\sin (a - b) + \sin (a + b)} \right]\).

1) Tính giá trị của các biểu thức: \(A = \cos \frac{{5\pi }}{{12}}\cos \frac{{7\pi }}{{12}}\); \(B = \cos {75^o}\sin {15^o}\).

Giải:

\(A = \cos \frac{{5\pi }}{{12}}\cos \frac{{7\pi }}{{12}} = \frac{1}{2}\left[ {\cos \left( {\frac{{5\pi }}{{12}} - \frac{{7\pi }}{{12}}} \right) + \cos \left( {\frac{{5\pi }}{{12}} + \frac{{7\pi }}{{12}}} \right)} \right]\)

\( = \frac{1}{2}\left[ {\cos \left( { - \frac{\pi }{6}} \right) + \cos \pi } \right] = \frac{1}{2}\left( {\frac{{\sqrt 3 }}{2} - 1} \right) = \frac{{\sqrt 3  - 2}}{4}\).

\(B = \cos {75^o}\sin {15^o} = \frac{1}{2}\left[ {\sin ({{15}^o} - {{75}^o}) + \sin ({{15}^o} + {{75}^o})} \right]\)

\( = \frac{1}{2}\left[ {\sin ( - {{60}^o}) + \sin {{90}^o}} \right] = \frac{1}{2}\left( { - \frac{{\sqrt 3 }}{2} + 1} \right) = \frac{{2 - \sqrt 3 }}{4}\).

2) Cho \(\sin 2x =  - \frac{1}{3}\). Tính \(\sin \left( {x + \frac{\pi }{4}} \right)\cos \left( {x - \frac{\pi }{4}} \right)\).

Giải:

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

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