Question

(i) We know that the range of the principal value of ( cos ^{-1} ) is ( [0, pi] )
Let ( cos ^{-1}left(frac{-1}{sqrt{2}}right)=theta ) Then
[
cos theta=frac{-1}{sqrt{2}}=-cos frac{pi}{4}=cos left(pi-frac{pi}{4}right)=cos frac{3 pi}{4}
]
( therefore quad theta=frac{3 pi}{4} in[0, pi] )
Hence, the principal value of ( cos ^{-1}left(frac{-1}{sqrt{2}}right) ) is ( frac{3 pi}{4} )
(ii) We know that the range of the principal value of ( cos ^{-1} ) is ( [0, pi] )
Let ( cos ^{-1}left(frac{-1}{2}right)= ) Q. Then
[
cos theta=frac{-1}{2}=-cos frac{pi}{3}=cos left(pi-frac{pi}{3}right)=cos frac{2 pi}{3}
]
( therefore quad theta=frac{2 pi}{3} in[0, pi] )
Hence, the principal value of ( cos ^{-1}left(frac{-1}{2}right) ) is ( frac{2 pi}{3} )
(iii) We know that the range of the principal value of ( cot ^{-1} ) is ( (0, pi) )
Let ( cot ^{-1}left(frac{-1}{sqrt{3}}right)=theta ). Then
( cot theta=frac{-1}{sqrt{3}}=-cot frac{pi}{3}=cot left(pi-frac{pi}{3}right)=cot frac{2 pi}{3} )
( therefore quad theta=frac{2 pi}{3} in(0, pi) )
Hence, the principal value of ( cot ^{-1}left(frac{-1}{sqrt{3}}right) ) is ( frac{2 pi}{3} )

# Find the principal value of each of the following: 6) cost () (ii) cos1 ) (ii) cot-

Solution