Question

известно,что cos a =3/5?0<a<п/2Вычислить:
а)tg(a+п/3)            б)tg(a-5п/4)

Answer 1

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Answer 2

$0<alpha <frac{pi }{2}Rightarrow Sinalpha>0\\Cosalpha=frac{3}{5}\\Sinalpha=sqrt{1-Cos^{2}alpha}=sqrt{1-(frac{3}{5})^{2}}=sqrt{1-frac{9}{25}}=sqrt{frac{16}{25}}=frac{4}{5}$

$tgalpha =frac{Sinalpha }{Cosalpha}=frac{4}{5}:frac{3}{5}=frac{4}{5}*frac{5}{3}=frac{4}{3}\\1)tg(alpha+frac{pi }{3})=frac{tgalpha+tgfrac{pi }{3}}{1-tgalpha*tgfrac{pi }{3}}=frac{frac{4}{3}+sqrt{3}}{1-frac{4}{3}*sqrt{3}}=frac{(4+3sqrt{3})*3} {3*(3-4sqrt{3})}=frac{4+3sqrt{3}}{3-4sqrt{3}}=frac{(4+3sqrt{3})(3+4sqrt{3})}{(3-4sqrt{3})(3+4sqrt{3})}=frac{12+16sqrt{3}+9sqrt{3}+36}{9-48}=-frac{48+25sqrt{3}}{39}$

$2)tg(alpha-frac{5pi }{4})=-tg(pi+(frac{pi }{4}-alpha))=-tg(frac{pi }{4}-alpha)=-frac{tgfrac{pi }{4}-tgalpha}{1+tgfrac{pi }{4}*tgalpha}=-frac{1-frac{4}{3}}{1+1*frac{4}{3}}=-frac{-frac{1}{3}}{frac{7}{3}}=frac{1}{7}$