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Question Number 30356 by soksan last updated on 21/Feb/18

$$\mathrm{If}\sin 2\theta =\cos 3\theta \mathrm{and}\theta \mathrm{is}\mathrm{an}\mathrm{acute}$$$$\mathrm{angle},\mathrm{then}\sin \theta \mathrm{equals}$$

Answered by mrW2 last updated on 21/Feb/18

$$\sin 2\theta =\cos 3\theta$$$$2\sin \theta \cos \theta ={4\cos }^3\theta -3\cos \theta$$$$2\sin \theta ={4\cos }^2\theta -3$$$$2\sin \theta =4(1-{\sin }^2\theta )-3$$$${4\sin }^2\theta +2\sin \theta -1=0$$$$\Rightarrow \sin \theta =\frac{-2+\sqrt{4+16}}{8}=\frac{\sqrt{5}-1}{4}$$$$or$$$$\sin 2\theta =\cos 3\theta$$$$\Rightarrow 2\theta +3\theta =90°$$$$\Rightarrow \theta =18°$$$$\Rightarrow \sin \theta =\frac{\sqrt{5}-1}{4}$$

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