how to find the exact value of sin(cos^-1(1/5)?

Draw a right triangle.

First, consider  cos^-1(1/5)

This means that the angle has a cos of 1/5   --->  the adjacent side = 1 and the hypotenuse = 5

So, label one of the sides 1 and the hypotenuse 5.

Now, use the Pythagorean Theorem to find the missing side (the opposite side):  1² + b²  =  5²  

--->   1 + b²  =  25   --->   b²  =  24   --->   b  =  √(24)  =  2√(6)

Finally, the sin will be the opposite side divided by the hypotenuse:  2√(6) / 5