Question

How do you find the exact value of `sin(cos^-1(sqrt5/5))`?

Answer 1

`sin(cos^-1(sqrt(5)/5))=(2sqrt(5))/5`

Explanation:

Let `cos^-1(sqrt(5)/5)=A` then `cosA=sqrt(5)/5`

and `sinA=sqrt(1-cos^2A)=sqrt(1-(sqrt(5)/5)^2)=(2sqrt(5))/5`
`rarrA=sin^-1((2sqrt(5))/5)`

Now, `sin(cos^-1(sqrt(5)/5))=sin(sin^-1((2sqrt(5))/5))=(2sqrt(5))/5`