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Giải bài 25 trang 70 sách bài tập toán 11 - Kết nối tri thức với cuộc sống

Cho \({\rm{sin}}x = - \frac{1}{3},x \in \left( {\pi ;\frac{{3\pi }}{2}} \right)\). Tính giá trị \({\rm{cos}}\left( {2x - \frac{\pi }{3}} \right)\).

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Đề bài

Cho \({\rm{sin}}x =  - \frac{1}{3},x \in \left( {\pi ;\frac{{3\pi }}{2}} \right)\). Tính giá trị \({\rm{cos}}\left( {2x - \frac{\pi }{3}} \right)\).

Phương pháp giải - Xem chi tiết

Xét dấu \(\cos x\) khi  \(x \in \left( {\pi ;\frac{{3\pi }}{2}} \right)\) Thay vào đẳng thức \({\sin ^2}x + {\cos ^2}x = 1\) ta tìm được \(\cos x\) Áp dụng công thức cộng \(\cos \left( {a - b} \right) = \cos a.\cos b + \sin a.\sin b\)

Lời giải chi tiết

\(x \in \left( {\pi ;\frac{{3\pi }}{2}} \right) \Rightarrow \cos x < 0 \Rightarrow \cos x =  - \sqrt {1 - {{\sin }^2}x}  =  - \sqrt {1 - {{\left( { - \frac{1}{3}} \right)}^2}}  =  - \frac{{2\sqrt 2 }}{3}\)Ta tính được: \({\rm{cos}}x =  - \frac{{2\sqrt 2 }}{3}\). Khi đó:\({\rm{cos}}\left( {2x - \frac{\pi }{3}} \right) = \frac{1}{2}{\rm{cos}}2x + \frac{{\sqrt 3 }}{2}{\rm{sin}}2x = \frac{1}{2}\left( {1 - 2{\rm{si}}{{\rm{n}}^2}x} \right) + \sqrt 3 {\rm{sin}}x{\rm{cos}}x = \frac{{7 + 4\sqrt 6 }}{{18}}\).

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