Question

Найдите Tg (а + П/4) , если cos2a=1/3 ; a э (0; П/2)

Answer 1

Cos 2a = 1/3
2cos²a - 1 =1/3
2cos²a = 4/3
cos²a = 2/3
tg²a + 1 = $\frac{1}{cosвa}$
tg²a = $\frac{1-cosвa}{cosвa}$
tga = $\sqrt{ \frac{1-cosвa}{cosвa}}= \sqrt{ \frac{ \frac{1}{3} }{ \frac{2}{3}} } = \sqrt{ \frac{1}{2} } = \frac{1}{ \sqrt{2} }$
tg(a + π/4) = $\frac{tga+tg \frac{pi}{4} }{1-tga*tg \frac{pi}{4}} = \frac{ \frac{1}{ \sqrt{2} } +1}{1- \frac{1}{ \sqrt{2}}} = \frac{ \frac{ \sqrt{2}+1}{\sqrt{2}} }{ \frac{ \sqrt{2}-1}{\sqrt{2}} } = \frac{ \sqrt{2}+1}{ \sqrt{2}-1} =$$\frac{2+2 \sqrt{2} +1}{ 2 -1в} =3+2 \sqrt{2}$