Question

Вычислите.......................................

Answer 1

$1); ; frac{tg frac{pi}{9}+tg frac{5pi}{36} }{1+tg frac{31pi}{36}cdot tg frac{pi}{9} } = frac{tg frac{pi}{9} +tg(pi - frac{31pi}{36}) }{1+tg frac{31pi}{36} cdot tgfrac{pi}{9} } = frac{tg frac{pi}{9} -tg frac{31pi}{36} }{1+tg frac{31pi}{36}cdot tg frac{pi}{9} } =tg( frac{pi }{9} -frac{31pi}{36} )=\\=tg(-frac{27pi}{36})=-tgfrac{27pi}{36}=-tg frac{3pi}{4}=-tg(pi -frac{pi}{4})=tg frac{pi}{4}=1$

$2); ; tgx=-4; ,; ; -90^circ textless x textless 90^circ \\tgx textless 0; ; i; ; -90^circ textless x textless 90^circ ; ; ; Rightarrow ; ; ; xin 4; chetverti; ; to ; ; sinx textless 0\\1+ctg^2x=frac{1}{sin^2x}; ,; ; to ; ; ; sin^2x=frac{1}{1+ctg^2x}= frac{1}{1+frac{1}{tg^2x}} = frac{tg^2x}{1+tg^2x} \\sin^2x= frac{(-4)^2}{1+(-4)^2} = frac{16}{1+16} = frac{16}{17} \\Tak; kak; ; sinx textless 0; ,; ; to; ; sinx=-sqrt{sin^2x}=-sqrt{ frac{16}{17}}=-frac{4}{sqrt{17}} =-frac{4sqrt{17}}{17}$