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

11th - 12th Class
Maths
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(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} )
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