[tex] \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 } )}}} [/tex]
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Question:-
[tex] \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 } )}.[/tex]
Given:-
[tex] \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 } )}.[/tex]
To Find:-
Solution:-
[tex] \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 } )}.[/tex]
[tex] \sf \large = cos(cos {}^{ - 1} ( \frac{3}{7} ) + sin(sin {}^{ - 1} ( \frac{2}{7} )) + cos(cos {}^{ - 1} ( \frac{6}{\sqrt{85} } ).[/tex]
[tex] \sf \large = \frac{3}{7} - \frac{2}{7} + \frac{6}{ \sqrt{85} } .[/tex]
[tex] \sf \large = \frac{1}{7} + \frac{6}{85} .[/tex]
Formula Used:-
[tex]\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} } ).[/tex]
Answer:-
[tex]{ \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 } )} =
= \frac{1}{7} + \frac{6}{ \sqrt{85} } }}[/tex]
Hope you have satisfied. ⚘
[tex]{\large{\underbrace{\mathbb{\pink{ ANSWER\: \:-: }}}}}[/tex]
Question:-
[tex] \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 } )}.[/tex]
Given:-
[tex] \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 } )}.[/tex]
To Find:-
[tex] \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 } )}.[/tex]
Solution:-
[tex] \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 } )}.[/tex]
[tex] \tt\large = cos(cos {}^{ - 1} ( \frac{3}{7} ) + sin(sin {}^{ - 1} ( \frac{2}{7} )) + cos(cos {}^{ - 1} ( \frac{6}{\sqrt{85} } ).[/tex]
[tex] \tt \large = \frac{3}{7} - \frac{2}{7} + \frac{6}{ \sqrt{85} } .[/tex]
[tex] \tt \large = \frac{1}{7} + \frac{6}{85} .[/tex]
Formula Used:-
[tex]\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} } ).[/tex]
Final Answer:-
[tex]{ \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 } )} =
= \frac{1}{7} + \frac{6}{\sqrt{85}}}}[/tex]