Question

$\mathtt \purple{ \mathtt{\boxed{Find \: the \: value : \sin ( \frac { \pi } { 2 } - \cos ^ { - 1 } \frac { 3 } { 7 } ) + \cos ( \frac { 3 \pi } { 2 } - \sin ^ { - 1 } \frac { 2 } { 7 } ) + \cos ( \tan ^ { - 1 } \frac { 7 } { 6 } )}}}$

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Answer 1

Question:-

$\sf \large{Find \: the \: value : sin ( \frac { \pi } { 2 } - cos ^ { - 1 } \frac { 3 } { 7 } ) + cos ( \frac { 3 \pi } { 2 } - sin ^ { - 1 } \frac { 2 } { 7 } ) + cos ( tan ^ { - 1 } \frac { 7 } { 6 } )}.$

Given:-

$\sf \large{ sin ( \frac { \pi } { 2 } - cos ^ { - 1 } \: \: \frac { 3 } { 7 } ) + cos ( \frac { 3 \pi } { 2 } - sin ^ { - 1 } \: \: \frac { 2 } { 7 } ) + cos ( tan ^ { - 1 } \: \: \frac { 7 } { 6 } )}.$

To Find:-

• $\sf \large Value \: of \: \sf \large{ sin ( \frac { \pi } { 2 } - cos ^ { - 1 } \frac { 3 } { 7 } ) + cos ( \frac { 3 \pi } { 2 } - sin ^ { - 1 } \frac { 2 } { 7 } ) + cos ( tan ^ { - 1 } \frac { 7 } { 6 } )}.$

Solution:-

$\sf \large{ sin ( \frac { \pi } { 2 } - cos ^ { - 1 } \: \: \frac { 3 } { 7 } ) + cos ( \frac { 3 \pi } { 2 } - sin ^ { - 1 } \: \: \frac { 2 } { 7 } ) + cos ( tan ^ { - 1 } \: \: \frac { 7 } { 6 } )}.$

$\sf \large = cos(cos {}^{ - 1} ( \frac{3}{7} ) + sin(sin {}^{ - 1} ( \frac{2}{7} )) + cos(cos {}^{ - 1} ( \frac{6}{\sqrt{85} } ).$

$\sf \large = \frac{3}{7} - \frac{2}{7} + \frac{6}{ \sqrt{85} } .$

$\sf \large = \frac{1}{7} + \frac{6}{85} .$

Formula Used:-

$\sf \large sin {}^{ - 1}(sin \: x) = x . \\ \\\sf \large cos {}^{ - 1} (cos \: x) = x. \\ \\ \sf \large tan \: x = \frac{7}{6} . \\ \\ \sf \large cos \: x = \frac{6}{ \sqrt{85} } = > x = cos {}^{ - 1} ( \frac{6}{ \sqrt{85} } ).$

Answer:-

${ \boxed{ \sf \large \color{yellow}{ sin ( \frac { \pi } { 2 } - cos ^ { - 1 } \: \: \frac { 3 } { 7 } ) + cos ( \frac { 3 \pi } { 2 } - sin ^ { - 1 } \: \: \frac { 2 } { 7 } ) + cos ( tan ^ { - 1 } \: \: \frac { 7 } { 6 } )} = </p><p> = \frac{1}{7} + \frac{6}{ \sqrt{85} } }}$

Hope you have satisfied. ⚘

Answer 2

${\large{\underbrace{\mathbb{\pink{ ANSWER\: \:-: }}}}}$

Question:-

$\tt \large{Find \: the \: value : sin ( \frac { \pi } { 2 } - cos ^ { - 1 } \frac { 3 } { 7 } ) + cos ( \frac { 3 \pi } { 2 } - sin ^ { - 1 } \frac { 2 } { 7 } ) + cos ( tan ^ { - 1 } \frac { 7 } { 6 } )}.$

Given:-

$\tt \large{ sin ( \frac { \pi } { 2 } - cos ^ { - 1 } \: \: \frac { 3 } { 7 } ) + cos ( \frac { 3 \pi } { 2 } - sin ^ { - 1 } \: \: \frac { 2 } { 7 } ) + cos ( tan ^ { - 1 } \: \: \frac { 7 } { 6 } )}.$

To Find:-

$\tt \large Value \: of \: \sf \large{ sin ( \frac { \pi } { 2 } - cos ^ { - 1 } \frac { 3 } { 7 } ) + cos ( \frac { 3 \pi } { 2 } - sin ^ { - 1 } \frac { 2 } { 7 } ) + cos ( tan ^ { - 1 } \frac { 7 } { 6 } )}.$

Solution:-

$\tt \large{ sin ( \frac { \pi } { 2 } - cos ^ { - 1 } \: \: \frac { 3 } { 7 } ) + cos ( \frac { 3 \pi } { 2 } - sin ^ { - 1 } \: \: \frac { 2 } { 7 } ) + cos ( tan ^ { - 1 } \: \: \frac { 7 } { 6 } )}.$

$\tt\large = cos(cos {}^{ - 1} ( \frac{3}{7} ) + sin(sin {}^{ - 1} ( \frac{2}{7} )) + cos(cos {}^{ - 1} ( \frac{6}{\sqrt{85} } ).$

$\tt \large = \frac{3}{7} - \frac{2}{7} + \frac{6}{ \sqrt{85} } .$

$\tt \large = \frac{1}{7} + \frac{6}{85} .$

Formula Used:-

$\tt \large sin {}^{ - 1}(sin \: x) = x . \\ \\\tt \large cos {}^{ - 1} (cos \: x) = x. \\ \\ \tt \large tan \: x = \frac{7}{6} . \\ \\ \tt \large cos \: x = \frac{6}{ \sqrt{85} } = > x = cos {}^{ - 1} ( \frac{6}{ \sqrt{85} } ).$

Final Answer:-

${ \boxed{ \rm \large \color{pink}{ sin ( \frac { \pi } { 2 } - cos ^ { - 1 } \: \: \frac { 3 } { 7 } ) + cos ( \frac { 3 \pi } { 2 } - sin ^ { - 1 } \: \: \frac { 2 } { 7 } ) + cos ( tan ^ { - 1 } \: \: \frac { 7 } { 6 } )} = </p><p> = \frac{1}{7} + \frac{6}{\sqrt{85}}}}$