Question Number 151742 by mnjuly1970 last updated on 22/Aug/21
$$\:\: \\ $$$$\:\:\:\mathrm{F}\:\left({x}\:\right):=\:\frac{{log}\:\left({sin}\left({x}\right)\:+{cos}\:\left({x}\right)\right)}{{log}\:\left({sin}\left(\mathrm{2}{x}\right)\right)} \\ $$$$\:\:{find}\:\:\:{the}\:{Domain}\:{of}\:\:\:\:\mathrm{F}\:… \\ $$$$\:\:\:\mathrm{D}_{\:\mathrm{F}} \:=? \\ $$
Answered by mindispower last updated on 22/Aug/21
![F(x+2π)=F(x) we can study f in [0,2π[ sin(2x)>0⇒x∈]0,(π/2)[ sin(x)+cos(x)=(√2)sin(x+(π/4))>0⇒x∈]−(π/4),((3π)/4)[ DF=]0,(π/2)[](https://www.tinkutara.com/question/Q151749.png)
$${F}\left({x}+\mathrm{2}\pi\right)={F}\left({x}\right) \\ $$$${we}\:{can}\:{study}\:{f}\:{in}\:\left[\mathrm{0},\mathrm{2}\pi\left[\right.\right. \\ $$$$\left.{sin}\left(\mathrm{2}{x}\right)>\mathrm{0}\Rightarrow{x}\in\right]\mathrm{0},\frac{\pi}{\mathrm{2}}\left[\right. \\ $$$$\left.{sin}\left({x}\right)+{cos}\left({x}\right)=\sqrt{\mathrm{2}}{sin}\left({x}+\frac{\pi}{\mathrm{4}}\right)>\mathrm{0}\Rightarrow{x}\in\right]−\frac{\pi}{\mathrm{4}},\frac{\mathrm{3}\pi}{\mathrm{4}}\left[\right. \\ $$$$\left.{DF}=\right]\mathrm{0},\frac{\pi}{\mathrm{2}}\left[\right. \\ $$