The value of 1π{216sin−1(sin7π6)+27cos−1(cos2π3)+28tan−1(tan5π4)+200cot−1(cot(−π4))} must be.

∵sin−1(sin7π6)=π+π6=−π6 cos−1(cos2π3)=2π3 tan−1(tan5π4)=5π4−π=π4 And cot−1{cot(−π4)}=π−π4=3π4 Hence, requires value is 1π{216×−π6+27×2π3+28×π4+2} =-36+18+7+150 =139

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The value of $\frac{1}{π} {216 \sin^{- 1} (\sin \frac{7 π}{6}) + 27 \cos^{- 1} (\cos \frac{2 π}{3}) + 28 \tan^{- 1} (\tan \frac{5 π}{4}) + 200 \cot^{- 1} (\cot (\frac{- π}{4}))}$ must be.

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answer is 139.

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Detailed Solution

$∵ \sin^{- 1} (\sin \frac{7 π}{6}) = π + \frac{π}{6} = - \frac{π}{6} \cos^{- 1} (\cos \frac{2 π}{3}) = \frac{2 π}{3} \tan^{- 1} (\tan \frac{5 π}{4}) = \frac{5 π}{4} - π = \frac{π}{4} A n d \cot^{- 1} {\cot (- \frac{π}{4})} = π - \frac{π}{4} = \frac{3 π}{4} H e n c e, r e q u i r e s v a l u e i s \frac{1}{π} {216 \times - \frac{π}{6} + 27 \times \frac{2 π}{3} + 28 \times \frac{π}{4} + 2} = - 36 + 18 + 7 + 150 = 139$