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Given \(\cot \theta=-\dfrac{12}{5}\) and \(\dfrac{\pi}{2}<\theta<\pi\), find the exact values of
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a. \(\cot \theta=-\dfrac{12}{7} \ \Rightarrow \ \tan \theta=-\dfrac{7}{12}\)
\(\text{Graphically, given} \ \ \dfrac{\pi}{2}<\theta<\pi:\)
\(x=\sqrt{7^2+12^2}=13\)
\(\sec \theta=-\dfrac{13}{12}\)
b. \(\sin \theta=\dfrac{7}{13}\)
Given \(\operatorname{cosec}\,\alpha=-\dfrac{17}{8}\) and \(\pi<\alpha<\dfrac{3 \pi}{2}\), find the exact values of
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a. \(\operatorname{cosec} \alpha=-\dfrac{17}{8} \ \Rightarrow \ \sin \alpha=-\dfrac{8}{17}\)
\(\text{Graphically, given} \ \ \pi<\alpha<\dfrac{3 \pi}{2}:\)
\(x=\sqrt{17^2-8^2}=15\)
\(\sec \alpha=-\dfrac{17}{15}\)
b. \(\tan \alpha=\dfrac{8}{15}\)
Given `sectheta = −37/12` for `0 < theta < pi`,
find the exact value of `text(cosec)\ theta`. (2 marks)
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`sec\ theta = -37/12 => cos\ theta = -12/37`
`text(Graphically:)`
`x=sqrt(37^2-12^2)=35`
`:.\ text(cosec)\ theta= 1/(sintheta)= 1/(35/37)= 37/35`
Given `cot theta = −24/7` for `−pi/2 < theta < pi/2`, find the exact value of
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a. `cot theta = −24/7\ \ =>\ \ tan theta=– 7/24`
`text(Graphically, given)\ −pi/2 < theta < pi/2:`
`x= sqrt(24^2 + 7^2)=25`
`sectheta= 1/(costheta)= 1/(24/25)= 25/24`
b. `sintheta = −7/25`