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Question Number 114122 by bemath last updated on 17/Sep/20

find the value sin (cos^(−1) ((3/5))+tan^(−1) ((7/(13))))

$${find}\:{the}\:{value}\:\mathrm{sin}\:\left(\mathrm{cos}^{−\mathrm{1}} \left(\frac{\mathrm{3}}{\mathrm{5}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{7}}{\mathrm{13}}\right)\right) \\ $$

Answered by mr W last updated on 17/Sep/20

tan^(−1) (7/(13))=sin^(−1) (7/( (√(218))))=cos^(−1) ((13)/( (√(218))))  cos^(−1) (3/5)=sin^(−1) (4/5)  sin (cos^(−1) ((3/5))+tan^(−1) ((7/(13))))  =(sin cos^(−1) (3/5)) (cos tan^(−1) (7/(13)))+cos (cos^(−1) (3/5)) sin (tan^(−1) (7/(13)))  =(sin sin^(−1) (4/5)) (cos cos^(−1) ((13)/( (√(218)))))+cos (cos^(−1) (3/5)) sin (sin^(−1) (7/( (√(218)))))  =(4/5)×((13)/( (√(218))))+(3/5)×(7/( (√(218))))  =((73)/( 5(√(218))))

$$\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{7}}{\mathrm{13}}=\mathrm{sin}^{−\mathrm{1}} \frac{\mathrm{7}}{\:\sqrt{\mathrm{218}}}=\mathrm{cos}^{−\mathrm{1}} \frac{\mathrm{13}}{\:\sqrt{\mathrm{218}}} \\ $$$$\mathrm{cos}^{−\mathrm{1}} \frac{\mathrm{3}}{\mathrm{5}}=\mathrm{sin}^{−\mathrm{1}} \frac{\mathrm{4}}{\mathrm{5}} \\ $$$$\mathrm{sin}\:\left(\mathrm{cos}^{−\mathrm{1}} \left(\frac{\mathrm{3}}{\mathrm{5}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{7}}{\mathrm{13}}\right)\right) \\ $$$$=\left(\mathrm{sin}\:\mathrm{cos}^{−\mathrm{1}} \frac{\mathrm{3}}{\mathrm{5}}\right)\:\left(\mathrm{cos}\:\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{7}}{\mathrm{13}}\right)+\mathrm{cos}\:\left(\mathrm{cos}^{−\mathrm{1}} \frac{\mathrm{3}}{\mathrm{5}}\right)\:\mathrm{sin}\:\left(\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{7}}{\mathrm{13}}\right) \\ $$$$=\left(\mathrm{sin}\:\mathrm{sin}^{−\mathrm{1}} \frac{\mathrm{4}}{\mathrm{5}}\right)\:\left(\mathrm{cos}\:\mathrm{cos}^{−\mathrm{1}} \frac{\mathrm{13}}{\:\sqrt{\mathrm{218}}}\right)+\mathrm{cos}\:\left(\mathrm{cos}^{−\mathrm{1}} \frac{\mathrm{3}}{\mathrm{5}}\right)\:\mathrm{sin}\:\left(\mathrm{sin}^{−\mathrm{1}} \frac{\mathrm{7}}{\:\sqrt{\mathrm{218}}}\right) \\ $$$$=\frac{\mathrm{4}}{\mathrm{5}}×\frac{\mathrm{13}}{\:\sqrt{\mathrm{218}}}+\frac{\mathrm{3}}{\mathrm{5}}×\frac{\mathrm{7}}{\:\sqrt{\mathrm{218}}} \\ $$$$=\frac{\mathrm{73}}{\:\mathrm{5}\sqrt{\mathrm{218}}} \\ $$

Commented by bemath last updated on 17/Sep/20

santuyy

$${santuyy} \\ $$

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