Question

Tính giá trị biểu thức sau:

\(H=cos\left(\frac{2\pi}{7}\right)+cos\left(\frac{4\pi}{7}\right)+cos\left(\frac{6\pi}{7}\right)\)

Chứng minh đẳng thức sau:

\(sinA+sinB+sinC=4cos\left(\frac{A}{2}\right)cos\left(\frac{B}{2}\right).cos\left(\frac{C}{2}\right)\). Biết A+B+C=pi

Answer 1

\(sin\left(\frac{\pi}{7}\right)H=sin\left(\frac{\pi}{7}\right)cos\left(\frac{2\pi}{7}\right)+sin\left(\frac{\pi}{7}\right)cos\left(\frac{4\pi}{7}\right)+sin\left(\frac{\pi}{7}\right)cos\left(\frac{6\pi}{7}\right)\)

\(=\frac{1}{2}\left[sin\left(\frac{3\pi}{7}\right)-sin\left(\frac{\pi}{7}\right)+sin\left(\frac{5\pi}{7}\right)-sin\left(\frac{3\pi}{7}\right)+sin\pi-sin\left(\frac{5\pi}{7}\right)\right]\)

\(=-\frac{1}{2}sin\left(\frac{\pi}{7}\right)\)

\(\Rightarrow H=-\frac{1}{2}\)

\(sinA+sinB+sinC=2sin\left(\frac{A+B}{2}\right)cos\left(\frac{A-B}{2}\right)+2sin\left(\frac{C}{2}\right)cos\left(\frac{C}{2}\right)\)

\(=2cos\frac{C}{2}cos\left(\frac{A-B}{2}\right)+2cos\left(\frac{A+B}{2}\right)cos\frac{C}{2}\)

\(=2cos\frac{C}{2}\left[cos\left(\frac{A-B}{2}\right)+cos\left(\frac{A+B}{2}\right)\right]\)

\(=4cos\frac{C}{2}cos\frac{A}{2}cos\frac{B}{2}\)