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Question Number 33186 by NECx last updated on 12/Apr/18

$$Findtheexactvalueofsin\theta if$$$$cos\theta =\frac{1}{57}and\theta isobtuse$$

Answered by MJS last updated on 12/Apr/18

$${\cos }^2\theta +{\sin }^2\theta =1$$$$\sin \theta =\pm \sqrt{1-{\cos }^2\theta }=\pm \sqrt{1-\frac{1}{{57}^2}}=$$$$=\pm \sqrt{\frac{{57}^2-1}{{57}^2}}=\pm \frac{4\sqrt{203}}{57}\Rightarrow$$$$\Rightarrow \theta \approx 89°\vee \theta \approx 271°$$$$\mathrm{since}\mathrm{both}\mathrm{are}\mathrm{not}\mathrm{obtuse}\mathrm{I}\mathrm{guess}$$$$\mathrm{it}\mathrm{should}\mathrm{be}\cos \theta =-\frac{1}{57}\mathrm{because}$$$$\mathrm{then}\theta \approx 91°\vee \theta \approx 269°$$$$\mathrm{and}\sin \theta =\frac{4\sqrt{203}}{57}$$

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