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Question Number 83473 by M±th+et£s last updated on 02/Mar/20

solve in R  sin(πln(x))+cos(πln(x))=1

$${solve}\:{in}\:{R} \\ $$$${sin}\left(\pi{ln}\left({x}\right)\right)+{cos}\left(\pi{ln}\left({x}\right)\right)=\mathrm{1} \\ $$

Answered by mind is power last updated on 02/Mar/20

⇒((√2)/2)sin(πln(x))+((√2)/2)cos(πln(x))=(1/(√2))  ⇒sin(πln(x)+(π/4))=(1/(√2))  ⇒πln(x)+(π/4)=(π/4)+2kπ  ∨((3π)/4)+2kπ  ⇒x=e^(2k) ∨x=e^((1/2)+2k) ,k∈Z

$$\Rightarrow\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}{sin}\left(\pi{ln}\left({x}\right)\right)+\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}{cos}\left(\pi{ln}\left({x}\right)\right)=\frac{\mathrm{1}}{\sqrt{\mathrm{2}}} \\ $$$$\Rightarrow{sin}\left(\pi{ln}\left({x}\right)+\frac{\pi}{\mathrm{4}}\right)=\frac{\mathrm{1}}{\sqrt{\mathrm{2}}} \\ $$$$\Rightarrow\pi{ln}\left({x}\right)+\frac{\pi}{\mathrm{4}}=\frac{\pi}{\mathrm{4}}+\mathrm{2}{k}\pi\:\:\vee\frac{\mathrm{3}\pi}{\mathrm{4}}+\mathrm{2}{k}\pi \\ $$$$\Rightarrow{x}={e}^{\mathrm{2}{k}} \vee{x}={e}^{\frac{\mathrm{1}}{\mathrm{2}}+\mathrm{2}{k}} ,{k}\in\mathbb{Z} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Commented by M±th+et£s last updated on 02/Mar/20

thank you sir

$${thank}\:{you}\:{sir} \\ $$

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