Mathematics

Problem

For any real number

x

satisfying

-1\le x\le 1

, consider the inverse trigonometric expressions

\arcsin x

and

\arccos x

. a. (i) State the principal value ranges of

\arcsin x

and

\arccos x

. Explain why these ranges are necessary for defining the inverse functions. b. (ii) Prove that

\arcsin x+\arccos x=\dfrac{\pi}{2}

for every

x

in the interval

[-1,1]

, making explicit reference to the principal value ranges obtained in part (i).