Question

1.cos105cos75 2.2cos7 30'cos37 303.sin82 30'cos37 30'найдите значение выражения

Answer 1

$1)\;cos105^ocos75^o=\frac{1}{2}(cos(105^o+75^o)+cos(105^o-75^o))=\\\frac{1}{2}(cos180^o+cos30^o)=\frac{1}{2}(-1+\frac{\sqrt3}{2})=\frac{\sqrt3}{4}-\frac{1}{2};\\\\ 2)\;2cos7^o30'cos37^o30'=\\ \frac{1}{2}*2(cos(7^o30'+37^o30')+cos(7^o30'-37^o30'))\\=cos45^o+cos30^o=\frac{\sqrt2}{2}+\frac{\sqrt3}{2}=\frac{\sqrt2+\sqrt3}{2};\\\\3)sin82^o30'cos37^o 30'=\\ \frac{1}{2}(sin(82^o30'+37^o30')+sin(82^o30'-37^o30'))=\\\frac{1}{2}(sin120^o+sin45^o)=\frac{1}{2}(\frac{\sqrt3}{2}+\frac{\sqrt2}{2})=\frac{\sqrt3+\sqrt2}{4}$