If-sin-P-8-17-where-3pi-2-lt-P-lt-2pi-and-cos-Q-24-25-where-pi-lt-Q-lt-3pi-2-what-is-the-value-of-cos-2P-Q-

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

If sin (P) = −(8/(17)) ,where ((3π)/2)<P<2π and cos (Q)=−((24)/(25)), where π<Q<((3π)/2) what is the value of cos (2P+Q) =?

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

Answered by bemath last updated on 04/Oct/20

(1) cos (P)= ((15)/(17)) ; sin (Q)=−(7/(25)) ⇒cos (2P+Q) = cos (2P).cos (Q)−sin (2P).sin (Q) =−((24)/(25))(1−2(((64)/(289))))−{2(−(8/(17)))(((15)/(17)))}.(−(7/(25))) = −((24)/(25))(((161)/(289)))−(((240)/(289))).((7/(25))) = −((5544)/(289×25)) = −((5544)/(7225))

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

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

sin P=−(8/(17)) ⇒cos P=((√(17^2 −8^2 ))/(17))=((15)/(17)) cos Q=−((24)/(25)) ⇒sin Q=−((√(25^2 −24^2 ))/(25))=−(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×((15^2 )/(17^2 ))−1)(−((24)/(25)))−2(−(8/(17))×((15)/(17)))(−(7/(25))) =−(((2×15^2 −17^2 )×24+2×8×15×7)/(17^2 ×25)) =−((5544)/(7225))

$$\mathrm{sin}\:{P}=−\frac{\mathrm{8}}{\mathrm{17}} \\ $$$$\Rightarrow\mathrm{cos}\:{P}=\frac{\sqrt{\mathrm{17}^{\mathrm{2}} −\mathrm{8}^{\mathrm{2}} }}{\mathrm{17}}=\frac{\mathrm{15}}{\mathrm{17}} \\ $$$$\mathrm{cos}\:{Q}=−\frac{\mathrm{24}}{\mathrm{25}} \\ $$$$\Rightarrow\mathrm{sin}\:{Q}=−\frac{\sqrt{\mathrm{25}^{\mathrm{2}} −\mathrm{24}^{\mathrm{2}} }}{\mathrm{25}}=−\frac{\mathrm{7}}{\mathrm{25}} \\ $$$$\mathrm{cos}\:\left(\mathrm{2}{P}+{Q}\right)=\mathrm{cos}\:\mathrm{2}{P}\:\mathrm{cos}\:{Q}−\mathrm{sin}\:\mathrm{2}{P}\:\mathrm{sin}\:{Q} \\ $$$$=\left(\mathrm{2}\:\mathrm{cos}^{\mathrm{2}} \:{P}−\mathrm{1}\right)\:\mathrm{cos}\:{Q}−\mathrm{2}\:\mathrm{sin}\:{P}\:\mathrm{cos}\:{P}\:\mathrm{sin}\:{Q} \\ $$$$=\left(\mathrm{2}×\frac{\mathrm{15}^{\mathrm{2}} }{\mathrm{17}^{\mathrm{2}} }−\mathrm{1}\right)\left(−\frac{\mathrm{24}}{\mathrm{25}}\right)−\mathrm{2}\left(−\frac{\mathrm{8}}{\mathrm{17}}×\frac{\mathrm{15}}{\mathrm{17}}\right)\left(−\frac{\mathrm{7}}{\mathrm{25}}\right) \\ $$$$=−\frac{\left(\mathrm{2}×\mathrm{15}^{\mathrm{2}} −\mathrm{17}^{\mathrm{2}} \right)×\mathrm{24}+\mathrm{2}×\mathrm{8}×\mathrm{15}×\mathrm{7}}{\mathrm{17}^{\mathrm{2}} ×\mathrm{25}} \\ $$$$=−\frac{\mathrm{5544}}{\mathrm{7225}} \\ $$