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Question Number 153483 by cherokeesay last updated on 07/Sep/21

Commented by mr W last updated on 07/Sep/21

$$thereisnouniqueanswer!$$$$examples:$$$$\sin x=\frac{3}{5},\cos y=\frac{4}{5}$$$$\Rightarrow \frac{1}{\sin x}+\frac{1}{\cos y}=\frac{5}{3}+\frac{5}{4}=\frac{35}{12}$$$$\sin x=\frac{2}{5},\cos y=1$$$$\Rightarrow \frac{1}{\sin x}+\frac{1}{\cos y}=\frac{5}{2}+1=\frac{7}{2}$$$$\sin x=\frac{7}{10},\cos y=\frac{7}{10}$$$$\Rightarrow \frac{1}{\sin x}+\frac{1}{\cos y}=\frac{10}{7}+\frac{10}{7}=\frac{20}{7}$$$$......$$$$$$$$youcouldasktofindtheminimum$$$$andmaximumof\frac{1}{\sin x}+\frac{1}{\cos y}.$$

Commented by mr W last updated on 07/Sep/21

$$\frac{20}{7}⩽\frac{1}{\sin x}+\frac{1}{\cos y}⩽\frac{7}{2}$$

Commented by yeti123 last updated on 08/Sep/21

$$\mathrm{the}\mathrm{question}\mathrm{is}\sin x+\cos y\mathrm{not}\sin x+\cos x.$$

Commented by cherokeesay last updated on 08/Sep/21

$$mysolution:$$$${(sinx+cosx)}^2=\frac{49}{25}\iff$$$${sin}^{2}x+{cos}^{2}x+2sinxcosx=\frac{49}{25}\Rightarrow$$$$2sinxcosx=sin2x=\frac{49}{25}-1=\frac{24}{25}$$$$\frac{1}{sinx}+\frac{1}{cosx}=\frac{sinx+cosy}{sinxcosx}=$$$$\frac{\frac{7}{5}}{\frac{1}{2}sin2x}=\frac{\frac{7}{5}}{\frac{1}{2}\times \frac{24}{25}}=\frac{\frac{7}{5}}{\frac{12}{25}}=\frac{7}{5}\times \frac{25}{12}=\frac{35}{12}$$$$$$

Commented by liberty last updated on 08/Sep/21

$$yourquestionis\sin x+\cos y$$$$not\sin x+\cos x$$

Commented by cherokeesay last updated on 08/Sep/21

$$errorthatIjustrectfied.$$$$thankyou$$

Answered by ybav last updated on 07/Sep/21

$$ithinkits25/49$$

Answered by maged last updated on 08/Sep/21

$$\mathrm{Solution}:$$$$\frac{1}{\sin x\cos x}(\sin x+\cos x)=\frac{7}{5\sin x\cos x}$$$$\Rightarrow \frac{1}{\sin x}+\frac{1}{\cos x}=\frac{7}{5\sin x\cos x}.....\left(1\right)$$$${(\sin x+\cos x)}^2={\left(\frac{7}{5}\right)}^2$$$$\Rightarrow 2\sin x\cos x+1=\frac{49}{25}$$$$\Rightarrow 2\sin x\cos x=\frac{49}{25}-1$$$$\Rightarrow 2\sin x\cos x=\frac{24}{25}$$$$\Rightarrow \sin x\cos x=\frac{12}{25}$$$$\frac{1}{\sin x\cos x}=\frac{25}{12}....\left(2\right)$$$$formeq\left(2\right)ineq\left(1\right)$$$$\frac{1}{\sin x}+\frac{1}{\cos x}=\frac{7}{5}\times \frac{25}{12}=\frac{35}{12}★$$$$$$

Commented by yeti123 last updated on 08/Sep/21

$$\mathrm{the}\mathrm{question}\mathrm{is}\sin x+\cos y\mathrm{not}\sin x+\cos x.$$

Commented by liberty last updated on 08/Sep/21

$$whyyoureplace\cos yby\cos x?$$

Commented by cherokeesay last updated on 08/Sep/21

$$errorthatIjustrectified=$$$$thanks$$

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