Menu Close

Find-the-exact-value-of-sin-if-cos-1-57-and-is-obtuse-




Question Number 33186 by NECx last updated on 12/Apr/18
Find the exact value of sinθ if  cosθ=(1/(57)) and θ is obtuse
$${Find}\:{the}\:{exact}\:{value}\:{of}\:{sin}\theta\:{if} \\ $$$${cos}\theta=\frac{\mathrm{1}}{\mathrm{57}}\:{and}\:\theta\:{is}\:{obtuse} \\ $$
Answered by MJS last updated on 12/Apr/18
cos^2 θ+sin^2 θ=1  sin θ=±(√(1−cos^2 θ))=±(√(1−(1/(57^2 ))))=  =±(√((57^2 −1)/(57^2 )))=±((4(√(203)))/(57)) ⇒  ⇒ θ≈89° ∨ θ≈271°  since both are not obtuse I guess  it should be cos θ=−(1/(57)) because  then θ≈91° ∨ θ≈269°  and sin θ=((4(√(203)))/(57))
$$\mathrm{cos}^{\mathrm{2}} \theta+\mathrm{sin}^{\mathrm{2}} \theta=\mathrm{1} \\ $$$$\mathrm{sin}\:\theta=\pm\sqrt{\mathrm{1}−\mathrm{cos}^{\mathrm{2}} \theta}=\pm\sqrt{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{57}^{\mathrm{2}} }}= \\ $$$$=\pm\sqrt{\frac{\mathrm{57}^{\mathrm{2}} −\mathrm{1}}{\mathrm{57}^{\mathrm{2}} }}=\pm\frac{\mathrm{4}\sqrt{\mathrm{203}}}{\mathrm{57}}\:\Rightarrow \\ $$$$\Rightarrow\:\theta\approx\mathrm{89}°\:\vee\:\theta\approx\mathrm{271}° \\ $$$$\mathrm{since}\:\mathrm{both}\:\mathrm{are}\:\mathrm{not}\:\mathrm{obtuse}\:\mathrm{I}\:\mathrm{guess} \\ $$$$\mathrm{it}\:\mathrm{should}\:\mathrm{be}\:\mathrm{cos}\:\theta=−\frac{\mathrm{1}}{\mathrm{57}}\:\mathrm{because} \\ $$$$\mathrm{then}\:\theta\approx\mathrm{91}°\:\vee\:\theta\approx\mathrm{269}° \\ $$$$\mathrm{and}\:\mathrm{sin}\:\theta=\frac{\mathrm{4}\sqrt{\mathrm{203}}}{\mathrm{57}} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *