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Question Number 13401 by Tinkutara last updated on 19/May/17

$$\mathrm{If}\sin \left(\pi \cos \theta \right)=\cos \left(\pi \sin \theta \right),\mathrm{then}$$$$\mathrm{prove}\mathrm{that}\sin 2\theta =\pm \frac{3}{4}$$

Answered by ajfour last updated on 19/May/17

$$\sin x=\cos y$$$$x=\frac{\pi }{2}\pm y\left(atleast\right)$$$$here\sin \left(\pi \cos \theta \right)=\cos \left(\pi \sin \theta \right)$$$$\pi \cos \theta =\frac{\pi }{2}+\pi \sin \theta ......\left(i\right)$$$$\pi \cos \theta =\frac{\pi }{2}-\pi \sin \theta ......\left(ii\right)$$$$from\left(i\right):$$$$\cos \theta -\sin \theta =\frac{1}{2},squaring$$$$1-\sin 2\theta =\frac{1}{4}\Rightarrow \sin 2\theta =\frac{3}{4};$$$$from\left(ii\right):$$$$\cos \theta +\sin \theta =\frac{1}{2},squaring$$$$1+\sin 2\theta =\frac{1}{4}\Rightarrow \sin 2\theta =-\frac{3}{4}.$$$$$$$$$$

Commented by Tinkutara last updated on 20/May/17

$$\mathrm{In}\left(\mathrm{i}\right){\mathrm{shouldn}}^{\prime }\mathrm{t}\mathrm{it}\mathrm{be}\mathrm{something}\mathrm{else}$$$$\mathrm{because}\cos (\frac{\pi }{2}+\theta )=-\sin \theta ?$$

Commented by ajfour last updated on 20/May/17

$$ihaveobtained+\frac{3}{4}aswell.$$

Commented by ajfour last updated on 20/May/17

$$if\sin A=\cos B$$$$weconcludeA=\frac{\pi }{2}+B,herein\left(i\right)$$

Commented by Tinkutara last updated on 20/May/17

$$\mathrm{Thanks}!$$

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