Question

Вычислить Tg$alpha/2$ если sin$alpha$ + cos$alpha$=$sqrt{7}$/2

Answer 1

$sina+cosa=frac{sqrt7}{2}\\frac{2tgfrac{a}{2}}{1+tg^2frac{a}{2}}+frac{1-tg^2frac{a}{2}}{1+tg^2frac{a}{2}}=frac{sqrt7}{2}\\t=tgfrac{a}{2}; ,; ; frac{2t}{1+t^2}+frac{1-t^2}{1+t^2}-frac{sqrt7}{2}=0; ; ,; ; frac{4t+2-2t^2-sqrt7-sqrt7t^2}{2(1+t^2)}=0; ,\\(-2-sqrt7)t^2+4t+(2-sqrt7)=0; ,\\(2+sqrt7)t^2-4t-(2-sqrt7)=0\\D/4=2^2+(2+sqrt7)(2-sqrt7)=4+4-7=1\\t_{1,2}=frac{2pm 1}{2+sqrt7}\\tgfrac{a}{2}=frac{2-1}{2+sqrt7}=frac{2-sqrt7}{(2+sqrt7)(2-sqrt7)}=frac{2-sqrt7}{4-7}=frac{2-sqrt7}{-3}=frac{sqrt7-2}{3}$

$tgfrac{a}{2}=frac{2+1}{2+sqrt7}=frac{3(2-sqrt7)}{(2+sqrt7)(2-sqrt7)}=frac{3(2-sqrt7)}{4-7}=frac{3(2-sqrt7)}{-3}=sqrt7-2$