Question

[Tuyết Phạm]

Tính giá trị biểu thức sau mà ko dùng máy tính

A = \(cos\frac{\pi}{7}cos\frac{3\pi}{7}cos\frac{5\pi}{7}\)

\(B=sin6^0sin42^0sin66^0sin78^0\)

\(C=cos\frac{x}{5}cos\frac{2x}{5}cos\frac{4x}{5}cos\frac{8x}{5}\)

\(D=sin\frac{x}{7}+2sin\frac{3x}{7}+sin\frac{5x}{7}\)

Answer 1

[Nguyễn Việt Lâm]

- Xét \(sin\frac{x}{5}=0\Rightarrow C=...\)

- Với \(sin\frac{x}{5}\ne0\)

\(C.sin\frac{x}{5}=sin\frac{x}{5}.cos\frac{x}{5}.cos\frac{2x}{5}cos\frac{4x}{5}cos\frac{8x}{5}\)

\(=\frac{1}{2}sin\frac{2x}{5}cos\frac{2x}{5}cos\frac{4x}{5}cos\frac{8x}{5}\)

\(=\frac{1}{4}sin\frac{4x}{5}cos\frac{4x}{5}cos\frac{8x}{5}=\frac{1}{8}sin\frac{8x}{5}cos\frac{8x}{5}\)

\(=\frac{1}{16}sin\frac{16x}{5}\Rightarrow C=\frac{sin\frac{16x}{5}}{16.sin\frac{x}{5}}\)

\(D=sin\frac{x}{7}+sin\frac{5x}{7}+2sin\frac{3x}{7}\)

\(=2sin\frac{3x}{7}cos\frac{2x}{7}+2sin\frac{3x}{7}\)

\(=2sin\frac{3x}{7}\left(cos\frac{2x}{7}+1\right)=4cos^2\frac{x}{7}.sin\frac{3x}{7}\)

Answer 2

[Nguyễn Việt Lâm]

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

\(\Rightarrow A.sin\frac{\pi}{7}=sin\frac{\pi}{7}.cos\frac{\pi}{7}.cos\frac{2\pi}{7}cos\frac{4\pi}{7}\)

\(=\frac{1}{2}sin\frac{2\pi}{7}cos\frac{2\pi}{7}cos\frac{4\pi}{7}=\frac{1}{4}sin\frac{4\pi}{7}cos\frac{4\pi}{7}\)

\(=\frac{1}{8}sin\frac{8\pi}{7}=\frac{1}{8}sin\left(\pi+\frac{\pi}{7}\right)=-\frac{1}{8}sin\frac{\pi}{7}\)

\(\Rightarrow A=-\frac{1}{8}\)

\(B=sin6.cos48.cos24.cos12\)

\(B.cos6=sin6.cos6.cos12.cos24.cos48\)

\(=\frac{1}{2}sin12.cos12.cos24.cos48=\frac{1}{4}sin24.cos24.cos48\)

\(=\frac{1}{8}sin48.cos48=\frac{1}{16}sin96\)

\(=\frac{1}{16}sin\left(90+6\right)=\frac{1}{16}cos6\Rightarrow B=\frac{1}{16}\)