Question

Find the value of tan "C) +cos (+ sin" (1).

11th - 12th Class

Maths

Solution

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SOLUTION We know that the ranges of principal values of ( tan ^{-1}, cos ^{-1} ) and [ sin ^{-1} text {are }left(frac{-pi}{2}, frac{pi}{2}right) ] ( [0, pi] ) and ( left[frac{-pi}{2}, frac{pi}{2}right] ) respectively Let ( tan ^{-1}(1)=theta_{1} . ) Then [ tan theta_{1}=1=tan frac{pi}{4} Rightarrow theta_{1}=frac{pi}{4} in[0, pi] ] Let ( cos ^{-1}left(frac{-1}{2}right)=theta_{2} . ) Then, [ begin{array}{ll} cos theta_{2}=frac{-1}{2}=-cos frac{pi}{3}=cos left(pi-frac{pi}{3}right)=cos frac{2 pi}{3} therefore quad theta_{2}=frac{2 pi}{3} in[0, pi] end{array} ] Let ( sin ^{-1}left(frac{-1}{2}right)=theta_{3} . ) Then, [ sin theta_{3}=frac{-1}{2}=-sin frac{pi}{6}=sin left(frac{-pi}{6}right) Rightarrow theta_{3}=frac{-pi}{6} inleft[frac{-pi}{2}, frac{pi}{2}right] ] [ therefore tan ^{-1}(1)+cos ^{-1}left(frac{-1}{2}right)+sin ^{-1}left(frac{-1}{2}right)=left(frac{pi}{4}+frac{2 pi}{3}-frac{pi}{6}right)=frac{3 pi}{4} ]