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Question Number 75846 by behi83417@gmail.com last updated on 18/Dec/19

$${\mathbf{sin}}^5\mathbf{x}+\sqrt{2}\mathbf{sinx}=1,\mathbf{x}\in [0,2\mathit{\pi }]$$

Answered by MJS last updated on 18/Dec/19

$$t=\sin x$$$$t^5+\sqrt{2}t-1=0$$$$\mathrm{only}\mathrm{approximation}\mathrm{possible}$$$$\mathrm{only}\mathrm{one}\mathrm{real}\mathrm{solution}$$$$t\approx .634429$$$$\Rightarrow x=.687270\vee x=2.45432$$

Commented by behi83417@gmail.com last updated on 18/Dec/19

$$\mathrm{thank}\mathrm{you}\mathrm{proph}:\mathrm{MJS}.$$$$\mathrm{this}\mathrm{is}\mathrm{my}\mathrm{typo}.\mathrm{the}\mathrm{original}\mathrm{question}\mathrm{is}:$$$$\mathbf{sin}5\mathbf{x}+\sqrt{2}\mathbf{sinx}=1.$$

Answered by MJS last updated on 18/Dec/19

$$\sin 5x+\sqrt{2}\sin x=1$$$$\sin 5x=({16\sin }^{4}x-{20\sin }^{2}x+5)\sin x$$$$\sin x=t$$$$16t^5-20t^3+(5+\sqrt{2})t=1$$$$t^5-\frac{5}{4}{t}^3+\frac{5+\sqrt{2}}{16}t-\frac{1}{16}=0$$$$\mathrm{approximation}$$$$t_1\approx .171177$$$$t_2\approx .551260$$$$t_3\approx .929365$$$$t_{4,5}\approx -.825901\pm .174831\mathrm{i}$$$$\Rightarrow$$$$x_{11}\approx .172024;x_{12}\approx 2.96957$$$$x_{21}\approx .583873;x_{22}\approx 2.55772$$$$x_{31}\approx 1.19269;x_{32}\approx 1.94890$$

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