Question Number 188721 by Rupesh123 last updated on 06/Mar/23
Answered by Frix last updated on 06/Mar/23
![x≥0 ⇒ 0≤(√x)<∞ −2.73581...≤sin x +2sin 2x ≤2.73581... [Exact values ±(((√(6(789+43(√(129))))))/(32))] ⇒ No possible solution for x>((3(789+43(√(129))))/(512))=7.48468... x_1 =0 x_2 ≈1.51276 x_3 ≈7.00698 x_4 ≈7.25028 No other real solutions](https://www.tinkutara.com/question/Q188724.png)
$${x}\geqslant\mathrm{0} \\ $$$$\Rightarrow \\ $$$$\mathrm{0}\leqslant\sqrt{{x}}<\infty \\ $$$$−\mathrm{2}.\mathrm{73581}…\leqslant\mathrm{sin}\:{x}\:+\mathrm{2sin}\:\mathrm{2}{x}\:\leqslant\mathrm{2}.\mathrm{73581}… \\ $$$$\:\:\:\:\:\:\:\:\:\:\left[\mathrm{Exact}\:\mathrm{values}\:\pm\frac{\left.\sqrt{\mathrm{6}\left(\mathrm{789}+\mathrm{43}\sqrt{\mathrm{129}}\right.}\right)}{\mathrm{32}}\right] \\ $$$$\Rightarrow\:\mathrm{No}\:\mathrm{possible}\:\mathrm{solution}\:\mathrm{for}\:{x}>\frac{\mathrm{3}\left(\mathrm{789}+\mathrm{43}\sqrt{\mathrm{129}}\right)}{\mathrm{512}}=\mathrm{7}.\mathrm{48468}… \\ $$$${x}_{\mathrm{1}} =\mathrm{0} \\ $$$${x}_{\mathrm{2}} \approx\mathrm{1}.\mathrm{51276} \\ $$$${x}_{\mathrm{3}} \approx\mathrm{7}.\mathrm{00698} \\ $$$${x}_{\mathrm{4}} \approx\mathrm{7}.\mathrm{25028} \\ $$$$\mathrm{No}\:\mathrm{other}\:\mathrm{real}\:\mathrm{solutions} \\ $$