Question

Fie x un numar real pentru care ctg(pi/4-x) = 2. Sa se calculeze tgx

Answer 1

$\it ctg(\dfrac{\pi}{4}-x) =2 \Rightarrow \dfrac{1}{ctg(\dfrac{\pi}{4}-x)} =\dfrac{1}{2} \Rightarrow tg(\dfrac{\pi}{4}-x)=\dfrac{1}{2}\Rightarrow \dfrac{tg\dfrac{\pi}{4}-tg x}{1+tg\dfrac{\pi}{4}tgx}=\dfrac{1}{2}\Rightarrow\\ \\ \\ \Rightarrow \dfrac{1-tgx}{1+tgx}=\dfrac{1}{2} \stackrel{derivare}{\Longrightarrow}\ \ \dfrac{1-tgx+1+tgx}{1+tgx}=\dfrac{1+2}{2} \Rightarrow \dfrac{2}{1+tgx}=\dfrac{3}{2} \Rightarrow \\ \\ \\ \Rightarrow3+3tgx=4 \Rightarrow tgx=\dfrac{1}{3}$

Answer 2

Răspuns:

tgx=1/3

Explicație pas cu pas:

tg(π/4-x)=1/2=(tg(π/4)-tgx)/(1+tg(π/4)tgx)

1/2=(1-tgx)/(1+tgx)

1+tgx=2-2tgx

tgx+2tgx=2-1

3tgx=1

tgx=1/3