Question

Find the values of the following trigonometric ratios:
(i) $\sin \frac{5\mathrm{\pi }}{3}$

(ii) sin 17π

(iii) $\tan \frac{11\mathrm{\pi }}{6}$

(iv) $\cos \left(-\frac{25\mathrm{\pi }}{4}\right)$

(v) $\tan \frac{7\pi }{4}$

(vi) $\sin \frac{17\pi }{6}$

(vii) $\cos \frac{19\pi }{6}$

(viii) $\sin \left(-\frac{11\pi }{6}\right)$

(ix) $\mathrm{cosec}\left(-\frac{20\pi }{3}\right)$

(x) $\tan \left(-\frac{13\pi }{4}\right)$

(xi) $\cos \frac{19\pi }{4}$

(xii) $\sin \frac{41\pi }{4}$

(xiii) $\cos \frac{39\pi }{4}$

(xiv) $\sin \frac{151\pi }{6}$

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\mathrm{We}\mathrm{have}: \\ \frac{5\mathrm{\pi }}{3}={\left(\frac{5}{3}\times 180\right)}^{°}=300°=\left(90°\times 3+30°\right) \\ 300°\mathrm{lies}\mathrm{in}\mathrm{the}\mathrm{fourth}\mathrm{quadrant}\mathrm{in}\mathrm{which}\mathrm{the}\mathrm{sine}\mathrm{function}\mathrm{is}\mathrm{negative}. \\ \mathrm{Also},3\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{integer}. \\ \therefore \sin \left(\frac{5\mathrm{\pi }}{3}\right)=\sin \left(300°\right)=\sin \left(90°\times 3+30°\right)=-\cos 30°=-\frac{\sqrt{3}}{2} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right)\mathrm{We}\mathrm{have}: \\ \sin 17\pi =\sin 3060° \\ 3060°=90°\times 34+0° \\ \mathrm{Clearly}3060°\mathrm{is}\mathrm{in}\mathrm{the}\mathrm{negative}\mathrm{direction}\mathrm{of}\mathrm{the}x-\mathrm{axis},\mathrm{i}.\mathrm{e}.\mathrm{on}\mathrm{the}\mathrm{boundary}\mathrm{line}\mathrm{of}\mathrm{the}\mathrm{II}\mathrm{and}\mathrm{III}\mathrm{quadrant}s. \\ \mathrm{Also},34\mathrm{is}\mathrm{an}\mathrm{even}\mathrm{integer}. \\ \therefore \sin \left(3060°\right)=\sin \left(90°\times 34+0°\right)=-\sin 0°=0 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iii}\right)\mathrm{We}\mathrm{have}: \\ \frac{11\mathrm{\pi }}{6}={\left(\frac{11}{6}\times 180\right)}^{°}=330°=\left(90°\times 3+60°\right) \\ 330°\mathrm{lies}\mathrm{in}\mathrm{the}\mathrm{fourth}\mathrm{quadrant}\mathrm{in}\mathrm{which}\mathrm{the}\mathrm{tangent}\mathrm{function}\mathrm{is}\mathrm{negative}. \\ \mathrm{Also},3\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{integer}. \\ \therefore \tan \left(\frac{11\mathrm{\pi }}{6}\right)=\tan \left(330°\right)=\tan \left(90°\times 3+60°\right)=-\cot 60°=-\frac{1}{\sqrt{3}} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iv}\right)\mathrm{We}\mathrm{have}: \\ \cos \left(-\frac{25\mathrm{\pi }}{4}\right)=\cos \left(1125°\right) \\ \cos \left(-1125°\right)=\cos \left(1125°\right)=\cos \left(90°\times 12+45°\right) \\ 1125°\mathrm{lies}\mathrm{in}\mathrm{the}\mathrm{first}\mathrm{quadrant}\mathrm{in}\mathrm{which}\mathrm{the}\mathrm{cosine}\mathrm{function}\mathrm{is}\mathrm{positive}. \\ \mathrm{Also},12\mathrm{is}\mathrm{an}\mathrm{even}\mathrm{integer}. \\ \therefore \cos \left(-1125°\right)=\cos \left(1125°\right)=\cos \left(90°\times 12+45°\right)=\cos 45°=\frac{1}{\sqrt{2}} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{v}\right)\mathrm{We}\mathrm{have}: \\ \tan \frac{7\pi }{4}=\tan 315° \\ 315°=\left(90°\times 3+45°\right) \\ 315°\mathrm{lies}\mathrm{in}\mathrm{the}\mathrm{fourth}\mathrm{quadrant}\mathrm{in}\mathrm{which}\mathrm{the}\mathrm{tangent}\mathrm{function}\mathrm{is}\mathrm{negative}. \\ \mathrm{Also},3\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{integer}. \\ \therefore \tan \left(315°\right)=\tan \left(90°\times 3+45°\right)=-\cot 45°=-1 \end{gathered}$$


$$\begin{gathered} \left(\mathrm{vi}\right)\mathrm{We}\mathrm{have}: \\ \sin \frac{17\pi }{6}=\sin 510° \\ 510°=\left(90°\times 5+60\right)° \\ 510°\mathrm{lies}\mathrm{in}\mathrm{the}\mathrm{second}\mathrm{quadrant}\mathrm{in}\mathrm{which}\mathrm{the}\mathrm{sine}\mathrm{function}\mathrm{is}\mathrm{positive}. \\ \mathrm{Also},5\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{integer}. \\ \therefore \sin \left(510°\right)=\sin \left(90°\times 5+60°\right)=\cos \left(60°\right)=\frac{1}{2} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{vii}\right)\mathrm{We}\mathrm{have}: \\ \cos \frac{19\pi }{6}=\cos 570° \\ 570°=\left(90°\times 6+30°\right) \\ 570°\mathrm{lies}\mathrm{in}\mathrm{the}\mathrm{third}\mathrm{quadrant}\mathrm{in}\mathrm{which}\mathrm{the}\mathrm{cosine}\mathrm{function}\mathrm{is}\mathrm{negative}. \\ \mathrm{Also},6\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{integer}. \\ \therefore \cos \left(570°\right)=\cos \left(90°\times 6+30°\right)=-\cos \left(30°\right)=-\frac{\sqrt{3}}{2} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{viii}\right)\mathrm{We}\mathrm{have}: \\ \sin \left(-\frac{11\pi }{6}\right)=\sin \left(-330°\right) \\ \sin \left(-330°\right)=-\sin \left(330°\right)=-\sin \left(90°\times 3+60°\right) \\ 330°\mathrm{lies}\mathrm{in}\mathrm{the}\mathrm{fourth}\mathrm{quadrant}\mathrm{in}\mathrm{which}\mathrm{the}\mathrm{sine}\mathrm{function}\mathrm{is}\mathrm{negative}. \\ \mathrm{Also},3\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{integer}. \\ \therefore \sin \left(-330°\right)=-\sin \left(330°\right)=-\sin \left(90°\times 3+60°\right)=-\left(-\cos 60°\right)=-\left(-\frac{1}{2}\right)=\frac{1}{2} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ix}\right)\mathrm{We}\mathrm{have}: \\ \mathrm{cosec}\left(-\frac{20\pi }{3}\right)=\mathrm{cosec}\left(-1200°\right) \\ \mathrm{cosec}\left(-1200°\right)=-\mathrm{cosec}\left(1200°\right)=-\mathrm{cosec}\left(90°\times 13+30°\right) \\ 1200°\mathrm{lies}\mathrm{in}\mathrm{the}\mathrm{second}\mathrm{quadrant}\mathrm{in}\mathrm{which}\mathrm{the}\mathrm{cosec}\mathrm{function}\mathrm{is}\mathrm{positive}. \\ \mathrm{Also},13\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{integer}. \\ \therefore \mathrm{cosec}\left(-1200°\right)=-\mathrm{cosec}\left(1200°\right)=-\mathrm{cosec}\left(90°\times 13+30°\right)=-\sec 30°=-\frac{2}{\sqrt{3}} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{x}\right)\mathrm{We}\mathrm{have}: \\ \tan \left(-\frac{13\pi }{4}\right)=\tan \left(-585°\right) \\ \tan \left(-585°\right)=-\tan \left(585°\right)=-\tan \left(90°\times 6+45°\right) \\ 585°\mathrm{lies}\mathrm{in}\mathrm{the}\mathrm{third}\mathrm{quadrant}\mathrm{in}\mathrm{which}\mathrm{tangent}\mathrm{function}\mathrm{is}\mathrm{positive}. \\ \mathrm{Also},6\mathrm{is}\mathrm{an}\mathrm{even}\mathrm{integer}. \\ \therefore \tan \left(-585°\right)=-\tan \left(585°\right)=-\tan \left(90°\times 6+45°\right)=-\tan 45°=-1 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{xi}\right)\mathrm{We}\mathrm{have}: \\ \cos \frac{19\pi }{4}=\cos 855° \\ 855°=90°\times 9+45° \\ 855°\mathrm{lies}\mathrm{in}\mathrm{the}\mathrm{second}\mathrm{quadrant}\mathrm{in}\mathrm{which}\mathrm{the}\mathrm{cosine}\mathrm{function}\mathrm{is}\mathrm{negative}. \\ \mathrm{Also},9\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{integer}. \\ \therefore \cos \left(855°\right)=\cos \left(90°\times 9+45°\right)=-\sin \left(45°\right)=-\frac{1}{\sqrt{2}} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{xii}\right)\mathrm{We}\mathrm{have}: \\ \sin \frac{41\pi }{4}=\sin 1845° \\ 1845°=90°\times 20+45° \\ 1845°\mathrm{lies}\mathrm{in}\mathrm{the}\mathrm{first}\mathrm{quadrant}\mathrm{in}\mathrm{which}\mathrm{the}\mathrm{sine}\mathrm{function}\mathrm{is}\mathrm{positive}. \\ \mathrm{Also},20\mathrm{is}\mathrm{an}\mathrm{even}\mathrm{integer}. \\ \therefore \sin \left(1845°\right)=\sin \left(90°\times 20+45°\right)=\sin \left(45°\right)=\frac{1}{\sqrt{2}} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{xiii}\right)\mathrm{We}\mathrm{have}: \\ \cos \frac{39\pi }{4}=\cos 1755° \\ 1755°=90°\times 19+45° \\ 1755°\mathrm{lies}\mathrm{in}\mathrm{the}\mathrm{fourth}\mathrm{quadrant}\mathrm{in}\mathrm{which}\mathrm{the}\mathrm{cosine}\mathrm{function}\mathrm{is}\mathrm{positive}. \\ \mathrm{Also},19\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{integer}. \\ \therefore \cos \left(1755°\right)=\cos \left(90°\times 19+45°\right)=\sin \left(45°\right)=\frac{1}{\sqrt{2}} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{xiv}\right)\mathrm{We}\mathrm{have}: \\ \sin \frac{151\pi }{6}=\sin 4530° \\ 4530°=90°\times 50+30° \\ 4530°\mathrm{lies}\mathrm{in}\mathrm{the}\mathrm{third}\mathrm{quadrant}\mathrm{in}\mathrm{which}\mathrm{the}\mathrm{sine}\mathrm{function}\mathrm{is}\mathrm{negative}. \\ \mathrm{Also},50\mathrm{is}\mathrm{an}\mathrm{even}\mathrm{integer}. \\ \therefore \sin \left(4530°\right)=\sin \left(90°\times 50+30°\right)=-\sin \left(30°\right)=-\frac{1}{2} \end{gathered}$$