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Question Number 164511 by HongKing last updated on 18/Jan/22

$$\sin \cdot \left(\pi \sin \mathrm{x}\right)-\cos \cdot \left(\pi \sin \mathrm{x}\right)=1$$$$\mathrm{find}\mathbf{x}=?$$

Answered by mindispower last updated on 18/Jan/22

$$sin\left(a\right)-cos\left(a\right)=\sqrt{2}sin(a-\frac{\pi }{4})$$$$\iff withea=\pi sin\left(x\right)$$$$sin(a-\frac{\pi }{4})=\frac{1}{\sqrt{2}}$$$$\iff a-\frac{\pi }{4}=\frac{\pi }{4}+2k\pi ,\frac{3\pi }{4}+2k\pi$$$$sin\left(x\right)=1+2k\in \{1,-1\}$$$$x=\frac{\pi }{2}+k\pi$$$$\pi sin\left(x\right)=\frac{\pi }{2}+2k\pi$$$$sin\left(x\right)=\frac{1}{2}+2k,\Rightarrow k=0$$$$sin\left(x\right)=\frac{1}{2},x\in \{\frac{\pi }{6}+2k\pi ,\frac{5\pi }{6}+2k\pi \}$$$$$$

Commented by HongKing last updated on 20/Jan/22

$$\mathrm{thank}\mathrm{you}\mathrm{dear}\mathrm{Sir}\mathrm{cool}$$

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