Question

tg(x+pi/4)=2. ( x apartine (0,pi/4). calculati sin x.

Answer 1

Salut,

$tg\left(x+\dfrac{\pi}4\right)=2\Leftrightarrow\dfrac{tgx+tg\left(\dfrac{\pi}4\right)}{1-tgx\cdot tg\left(\dfrac{\pi}4\right)}=2\Leftrightarrow\dfrac{tgx+1}{1-tgx\cdot 1}=2\Leftrightarrow\\\\\Leftrightarrow tgx+1=2-2\cdot tgx\Rightarrow 3\cdot tgx=1,\ deci\ tgx=\dfrac{1}3.\\\\sinx=\pm\dfrac{tgx}{\sqrt{1+tg^2x}}=\pm\dfrac{\dfrac{1}3}{\sqrt{1+\left(\dfrac{1}3\right)^2}}=\pm\dfrac{\dfrac{1}3}{\sqrt{\dfrac{10}9}}=\pm\dfrac{1}{\sqrt{10}}=\pm\dfrac{\sqrt{10}}{10}.\\\\Cum\ x\in\left(0,\ \dfrac{\pi}4 \right )\Rightarrow sinx>0,\ deci\ avem\ c\breve{a}\ sinx=+\dfrac{\sqrt{10}}{10}.$

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Green eyes.