Question

вычеслить
1) cos (arcsin 2/9)
2)sin (arccos 3/4)
3),sin (arctg 3)
4)cos (arcctg (-2))
5) tg ( arcsin 1/5)
6)
ctg (arctg 6)

Answer 1

$1)arcsin frac{2}{9}=alpha to sin alpha =sin(arcsin frac{2}{9} )=frac{2}{9}\cos(arcsinfrac{2}{9})=cos alpha =sqrt{1-sin^2 alpha }=sqrt{1-frac{4}{81}}=frac{sqrt{77}}{9}\2)arccocfrac{3}{4}= alpha to cos alpha =frac{3}{4}\sin(arccosfrac{3}{4})=sin alpha =sqrt{1-cos^2 alpha }=sqrt{1-frac{9}{16}}=frac{sqrt{7}}{4}\3)arctg3= alpha to tg alpha =3\1+tg^2 alpha =frac{1}{cos^2 alpha } to cos^2 alpha =frac{1}{1+tg^2 alpha }=frac{1}{1+9}=frac{1}{10}$
$sin alpha =sqrt{1-cos^2 alpha }=sqrt{1-frac{1}{10}}=frac{3}{sqrt{10}}=sin(arctg3)\4)cos(arcctg(-2))=cos alpha , ctg alpha =-2,\1+ctg^2 alpha =frac{1}{sin^2 alpha }to sin^2 alpha =frac{1}{1+ctg^2 alpha }=frac{1}{1+4}=frac{1}{5}\cos alpha =sqrt{1-sin^2 alpha} =sqrt{1-frac{1}{5}}=frac{2}{sqrt{5}}=cos(arcctg(-2))\5)tg(arcsinfrac{1}{5})=tg alpha ,to sin alpha =frac{1}{5}\cos alpha =sqrt{1-frac{1}{25}}=frac{sqrt{24}}{5}\tg alpha =frac{sin alpha }{cos alpha }$
$tg alpha =frac{1}{5}:frac{sqrt{24}}{5}=frac{1}{sqrt{24}}=tg(arcsinfrac{1}{5})\6)ctg(arctg6)=ctg alpha , alpha =arctg6,to tg alpha =6\ctg alpha =frac{1}{tg alpha }=frac{1}{6}$