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Evaluate the following inverse-trigonometric expressions: (i) sin((1/2) cos^(-1)(4/5)); (ii) sin^(-1)[cos(sin^(-1)(sqrt(3)/2))]; (iii) sin((1/4) arcsin(sqrt(63)/8)). Which option gives the three values in order?

1. (i) 1/sqrt(10) (ii) pi/6 (iii) 1/(2 sqrt(2))
2. (i) 1/sqrt(10) (ii) pi/3 (iii) 1/2
3. (i) 3/sqrt(10) (ii) pi/6 (iii) 1/(2 sqrt(2))
4. (i) 1/sqrt(5) (ii) pi/6 (iii) 1/4

Correct answer: (i) 1/sqrt(10) (ii) pi/6 (iii) 1/(2 sqrt(2))

Solution

Each part uses half-angle identities and the composition of inverse functions. Convert the inner inverse value to an angle, then apply the half/quarter-angle formula.

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