Question

Упростите выражение$frac{tg frac{ pi }{15}+tg frac{4 pi }{15} }{1-tg frac{4 pi }{15}*tg frac{ pi }{15} }$:

Ответ должен получиться 1,732

Answer 1

$frac{tgfrac{pi}{15}+tgfrac{4pi}{15}}{1-tgfrac{4pi}{15}bullet tgfrac{pi}{15}}=frac{frac{sin(frac{pi+4pi}{15})}{cosfrac{pi}{15}cosfrac{4 pi }{15}}}{1-frac{sinfrac{4pi}{15}}{cosfrac{4pi}{15}}bullet frac{sinfrac{pi}{15}}{cosfrac{pi}{15}}}=frac{sinfrac{pi}{3}}{cosfrac{pi}{15}cosfrac{4 pi }{15}(1-frac{sinfrac{4pi}{15}sinfrac{pi}{15}}{cosfrac{4pi}{15}cosfrac{pi}{15}})}=$

$=frac{frac{sqrt3}{2}}{cosfrac{pi}{15}cosfrac{4 pi }{15}-sinfrac{4pi}{15}sinfrac{pi}{15}}=frac{sqrt3}{2(frac{1}{2}(cosfrac{-3pi}{15}+cosfrac{5pi}{15})-frac{1}{2}(\cosfrac{-3pi}{15}-cosfrac{5pi}{15}))}=\\=frac{sqrt3}{cosfrac{-3pi}{15}+cosfrac{5pi}{15}-cosfrac{-3pi}{15}+cosfrac{5pi}{15}}=frac{sqrt3}{cosfrac{pi}{5}+cosfrac{pi}{3}-cosfrac{pi}{5}+cosfrac{pi}{3}}=frac{sqrt3}{frac{1}{2}+frac{1}{2}}=frac{sqrt3}{1}=\\=sqrt3.\\sqrt3approx1.732$