Question

How do you find `cos(1/2sin^-1(sqrt3/2))`?

Answer 1

`cos(1/2 sin^-1(sqrt3/2))=sqrt3/2`

Explanation:

`cos(1/2 sin^-1(sqrt3/2))`

The restriction for the range of arcsin x is `[-pi/2,pi/2]`. Since the argument is positive it means that our triangle is in quadrant I with the opposite side of `sqrt 3` and hypotenuse 2 and therefore the adjacent is 1. Note that we do't need to know the angle even though we can tell what it is. Let's just call the angle `theta` then we need to find `cos (1/2 theta)`. So we use the formula `cos (1/2theta)=sqrt(1/2(1+costheta)` to evaluate `cos (1/2 theta)`. That is,

`cos (1/2 theta)=sqrt(1/2(1+costheta)`

`=sqrt(1/2(1+1/2)`

`=sqrt(1/2(3/2)`

`=sqrt(3/4)=sqrt3/2`