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Question Number 13835 by Tinkutara last updated on 24/May/17

Commented by Tinkutara last updated on 24/May/17

$$\mathrm{Answer}\mathrm{given}\mathrm{is}2n\pi -\frac{3\pi }{4}<\frac{A}{2}<2n\pi -\frac{\pi }{4}$$

Answered by ajfour last updated on 24/May/17

$$ifA=\frac{\pi }{2}-2B,then\frac{A}{2}=\frac{\pi }{4}-B$$$$2\sin (\frac{\pi }{4}-B)=-\sqrt{1+\cos 2B}-\sqrt{1-\cos 2B}$$$$2\sin (\frac{\pi }{4}-B)=-\sqrt{2}\mid \cos B\mid -\sqrt{2}\mid \sin B\mid \mid$$$$\cos B-\sin B+\mid \cos B\mid +\mid \sin B\mid =0$$$$\Rightarrow \sin B⩾0,\cos B⩽0$$$$-(2n-1)\pi -\frac{\pi }{2}⩽B⩽-(2n-1)\pi$$$$-(2n-1)\pi -\frac{\pi }{2}⩽\frac{\pi }{4}-\frac{A}{2}⩽-(2n-1)\pi$$$$\Rightarrow \frac{A}{2}⩽2n\pi -\pi +\frac{\pi }{4}+\frac{\pi }{2}$$$$and\frac{A}{2}⩾2n\pi -\pi +\frac{\pi }{4}$$$$2\mathit{n}\mathit{\pi }-\frac{3\mathit{\pi }}{4}⩽\frac{A}{2}⩽2\mathit{n}\mathit{\pi }-\frac{\mathit{\pi }}{4}.$$

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