Question and Answers Forum

All Questions      Topic List

Trigonometry Questions

Previous in All Question      Next in All Question      

Previous in Trigonometry      Next in Trigonometry      

Question Number 15298 by Tinkutara last updated on 09/Jun/17

$$\mathrm{Solve}\mathrm{for}x\mathrm{and}y$$$$x^2+2x\sin \left(xy\right)+1=0$$

Commented by prakash jain last updated on 10/Jun/17

$$xy=u$$$$x^2+2x\sin \left(u\right)+1=0$$$$\sin u=-\frac{1+x^2}{2x}$$$$f\left(x\right)=-\frac{1+x^2}{2x}$$$$f\left(x\right)⩽-1orf\left(x\right)⩾1$$$$only2solutionforxforf\left(x\right)=\pm 1$$$$f\left(x\right)=-1\Rightarrow x=1$$$$f\left(x\right)=1\Rightarrow x=-1$$$$x=1$$$$\sin xy=-\frac{1+x^2}{2x}=-1$$$$\sin \left(y\right)=-1\Rightarrow y=-\frac{\pi }{2}$$$$x=-1$$$$\sin (-y)=-\frac{1+x^2}{2x}$$$$\sin (-y)=1$$$$\sin \left(y\right)=-1\Rightarrow \sin y=-1\Rightarrow y=-\frac{\pi }{2}$$$$\mathrm{solutions}:$$$$x=1,y=-\frac{\pi }{2}(ycantake2n\pi +\frac{3\pi }{2})$$$$x=-1,y=-\frac{\pi }{2}(ycantake2n\pi +\frac{3\pi }{2})$$

Commented by Tinkutara last updated on 10/Jun/17

$$\mathrm{But}\mathrm{answer}\mathrm{is}(x,y)=$$$$(1,(2n+1)\frac{3\pi }{2})\mathrm{and}(-1,(2n+1)\frac{\pi }{2})$$

Commented by prakash jain last updated on 10/Jun/17

$$x^2+2x\sin \left(xy\right)+1=0$$$$putx=1$$$$2\sin y+2=0$$$$\sin y=-1$$$$whatisthesolutionfory.$$$$compareitwithur{book}^{\prime }sanswer.$$

Commented by prakash jain last updated on 10/Jun/17

$$3n\pi +\frac{3\pi }{2}$$$$n=0y=\frac{3\pi }{2}$$$$n=1y=3\pi +\frac{3\pi }{2}=3\pi +\pi +\frac{\pi }{2}=4\pi +\frac{\pi }{2}$$$$putx=1,y=4\pi +\frac{\pi }{2}\mathrm{in}\mathrm{original}\mathrm{equation}$$$$\mathrm{does}\mathrm{it}\mathrm{satisfy}\mathrm{the}\mathrm{original}\mathrm{equation}?$$

Answered by Tinkutara last updated on 09/Jul/17

$$\mathrm{Given}\mathrm{equation}\mathrm{can}\mathrm{be}\mathrm{written}\mathrm{as}$$$${[x+\sin \left(xy\right)]}^2+{\cos }^2\left(xy\right)=0$$$$\therefore \cos \left(xy\right)=0\mathrm{and}x+\sin \left(xy\right)=0.$$$$\Rightarrow x=\pm 1$$$$\cos y=0;\sin y=-1\Rightarrow y=(2n+1)\frac{3\pi }{2}$$$$\Rightarrow (x,y)=(\pm 1,(2n+1)\frac{3\pi }{2})$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com