Tính giá trị của các biểu thức sau: a) A = sin pi /15cos pi /10 + sin pi /10cos pi /15cos 2pi /15cos pi /5 - sin 2pi /15sin pi /5; b) B =sin pi /32cos pi /32cos pi /16cos pi /8
33
24/07/2024
Tính giá trị của các biểu thức sau:
a) \(A = \frac{{\sin \frac{\pi }{{15}}\cos \frac{\pi }{{10}} + \sin \frac{\pi }{{10}}\cos \frac{\pi }{{15}}}}{{\cos \frac{{2\pi }}{{15}}\cos \frac{\pi }{5} - \sin \frac{{2\pi }}{{15}}\sin \frac{\pi }{5}}}\);
b) \(B = \sin \frac{\pi }{{32}}\cos \frac{\pi }{{32}}\cos \frac{\pi }{{16}}\cos \frac{\pi }{8}\).
Trả lời
Lời giải:
a) Ta có:
\(A = \frac{{\sin \frac{\pi }{{15}}\cos \frac{\pi }{{10}} + \sin \frac{\pi }{{10}}\cos \frac{\pi }{{15}}}}{{\cos \frac{{2\pi }}{{15}}\cos \frac{\pi }{5} - \sin \frac{{2\pi }}{{15}}\sin \frac{\pi }{5}}}\)\( = \frac{{\sin \frac{\pi }{{15}}\cos \frac{\pi }{{10}} + \cos \frac{\pi }{{15}}\sin \frac{\pi }{{10}}}}{{\cos \frac{{2\pi }}{{15}}\cos \frac{\pi }{5} - \sin \frac{{2\pi }}{{15}}\sin \frac{\pi }{5}}}\)
\( = \frac{{\sin \left( {\frac{\pi }{{15}} + \frac{\pi }{{10}}} \right)}}{{\cos \left( {\frac{{2\pi }}{{15}} + \frac{\pi }{5}} \right)}}\)\( = \frac{{\sin \frac{\pi }{6}}}{{\cos \frac{\pi }{3}}} = \frac{{\frac{1}{2}}}{{\frac{1}{2}}} = 1\).
b) Ta có:
\(B = \sin \frac{\pi }{{32}}\cos \frac{\pi }{{32}}\cos \frac{\pi }{{16}}\cos \frac{\pi }{8}\)\( = \left( {\frac{1}{2}.2\sin \frac{\pi }{{32}}\cos \frac{\pi }{{32}}} \right)\cos \frac{\pi }{{16}}\cos \frac{\pi }{8}\)
\( = \frac{1}{2}\sin \left( {2.\frac{\pi }{{32}}} \right)\cos \frac{\pi }{{16}}\cos \frac{\pi }{8}\)\( = \frac{1}{2}\sin \frac{\pi }{{16}}\cos \frac{\pi }{{16}}\cos \frac{\pi }{8}\)
\( = \frac{1}{4}.2\sin \frac{\pi }{{16}}\cos \frac{\pi }{{16}}\cos \frac{\pi }{8}\)\( = \frac{1}{4}\sin \frac{\pi }{8}\cos \frac{\pi }{8} = \frac{1}{8}.2\sin \frac{\pi }{8}\cos \frac{\pi }{8}\)
\( = \frac{1}{8}\sin \frac{\pi }{4} = \frac{1}{8}.\frac{{\sqrt 2 }}{2} = \frac{{\sqrt 2 }}{{16}}\).
