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Question Number 116448 by bobhans last updated on 04/Oct/20

$$\mathrm{If}\sin \left(\mathrm{P}\right)=-\frac{8}{17},\mathrm{where}\frac{3\pi }{2}<\mathrm{P}<2\pi$$ $$\mathrm{and}\cos \left(\mathrm{Q}\right)=-\frac{24}{25},\mathrm{where}\pi <\mathrm{Q}<\frac{3\pi }{2}$$ $$\mathrm{what}\mathrm{is}\mathrm{the}\mathrm{value}\mathrm{of}\cos (2\mathrm{P}+\mathrm{Q})=?$$

Answered by bemath last updated on 04/Oct/20

$$\left(1\right)\cos \left(\mathrm{P}\right)=\frac{15}{17};\sin \left(\mathrm{Q}\right)=-\frac{7}{25}$$ $$\Rightarrow \cos (2\mathrm{P}+\mathrm{Q})=\cos \left(2\mathrm{P}\right).\cos \left(\mathrm{Q}\right)-\sin \left(2\mathrm{P}\right).\sin \left(\mathrm{Q}\right)$$ $$=-\frac{24}{25}(1-2\left(\frac{64}{289}\right))-\left\{2(-\frac{8}{17})\left(\frac{15}{17}\right)\right\}.(-\frac{7}{25})$$ $$=-\frac{24}{25}\left(\frac{161}{289}\right)-\left(\frac{240}{289}\right).\left(\frac{7}{25}\right)$$ $$=-\frac{5544}{289\times 25}=-\frac{5544}{7225}$$

Answered by mr W last updated on 04/Oct/20

$$\sin P=-\frac{8}{17}$$ $$\Rightarrow \cos P=\frac{\sqrt{{17}^2-8^2}}{17}=\frac{15}{17}$$ $$\cos Q=-\frac{24}{25}$$ $$\Rightarrow \sin Q=-\frac{\sqrt{{25}^2-{24}^2}}{25}=-\frac{7}{25}$$ $$\cos (2P+Q)=\cos 2P\cos Q-\sin 2P\sin Q$$ $$=(2{\cos }^{2}P-1)\cos Q-2\sin P\cos P\sin Q$$ $$=(2\times \frac{{15}^2}{{17}^2}-1)(-\frac{24}{25})-2(-\frac{8}{17}\times \frac{15}{17})(-\frac{7}{25})$$ $$=-\frac{(2\times {15}^2-{17}^2)\times 24+2\times 8\times 15\times 7}{{17}^2\times 25}$$ $$=-\frac{5544}{7225}$$

Commented byI want to learn more last updated on 04/Oct/20

$$\mathrm{Sweet}\mathrm{sir}$$

Commented byI want to learn more last updated on 04/Oct/20

Commented byI want to learn more last updated on 04/Oct/20

$$\mathrm{Sir}\mathrm{please}\mathrm{tag}\mathrm{this}\mathrm{your}\mathrm{solution}.\mathrm{I}\mathrm{can}\mathrm{only}\mathrm{see}\mathrm{the}\mathrm{image}.\mathrm{but}\mathrm{i}\mathrm{think}\mathrm{it}\mathrm{is}$$ $$\mathrm{your}\mathrm{solution}\mathrm{sir}$$

Commented bymr W last updated on 04/Oct/20

$$whatisthequestion?$$

Commented byI want to learn more last updated on 04/Oct/20

$$\mathrm{I}\mathrm{only}\mathrm{see}\mathrm{the}\mathrm{image}\mathrm{saved}\mathrm{sir},\mathrm{no}\mathrm{question}\mathrm{number},\mathrm{am}\mathrm{only}\mathrm{thinking}\mathrm{it}$$ $$\mathrm{could}\mathrm{be}\mathrm{find}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{small}\mathrm{circle}.\mathrm{with}\mathrm{the}\mathrm{lines}\mathrm{you}\mathrm{drew}\mathrm{on}\mathrm{the}$$ $$\mathrm{diagram}$$

Commented bymr W last updated on 04/Oct/20

$$seeQ116466$$

Commented byI want to learn more last updated on 04/Oct/20

$$\mathrm{Am}\mathrm{really}\mathrm{grateful}\mathrm{sir}$$

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