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Question Number 63693 by Rio Michael last updated on 07/Jul/19

$$findthegeneralsolutionfor$$$$sin5\theta +sin3\theta =1$$

Commented by Prithwish sen last updated on 07/Jul/19

$$\sin 5\theta +\sin 3\theta =1$$$$(5\sin \theta -{20\sin }^3\theta +{16\sin }^5\theta )+(3\sin \theta -{4\sin }^3\theta )=1$$$$\Rightarrow {16\sin }^5\theta -{24\sin }^3\theta +8\sin \theta -1=0$$$$\mathrm{It}\mathrm{is}\mathrm{an}\mathrm{equation}\mathrm{of}{5}^{\mathrm{th}}\mathrm{degree}.\mathrm{I}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{think}$$$$\mathrm{it}\mathrm{is}\mathrm{an}\mathrm{easy}\mathrm{one}\mathrm{to}\mathrm{solve}.$$

Commented by Rio Michael last updated on 07/Jul/19

$$reallyithinkigotadifficultyinit$$

Answered by MJS last updated on 07/Jul/19

$$\mathrm{no}\mathrm{exact}\mathrm{solution}\mathrm{possible}$$$$\mathrm{for}x\in [0,2\pi [\mathrm{I}\mathrm{get}$$$$x_1\approx .132171549(\approx 7.57287194°)$$$$x_2\approx .620008878(\approx 35.5238919°)$$$$x_3=\pi -x_2\approx 2.52158378(\approx 144.476108°)$$$$x_4=\pi -x_1\approx 3.00942110(\approx 172.427128°)$$$$\Rightarrow \mathrm{generally}\underset{i=1}{\stackrel{4}{x_i}}+2n\pi \wedge n\in \mathbb{Z}(=\underset{i=1}{\stackrel{4}{x_i}}°+n\times 360°\wedge n\in \mathbb{Z})$$

Commented by Prithwish sen last updated on 08/Jul/19

$$\mathrm{Thank}\mathrm{you}\mathrm{sir}$$

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