To find an exact expression for sin(pi/5) using double-angle formulas, we can make use of the identity for sin(2x), where x = pi/10.
The double-angle formula for sine states:
sin(2x) = 2sin(x)cos(x)
Using the given value sin(pi/10) = (sqrt(5)-1)/4, we can substitute this into the double-angle formula:
sin(2(pi/10)) = 2sin(pi/10)cos(pi/10)
sin(pi/5) = 2 *(sqrt(5)-1)/4 * cos(pi/10)
Now, we need to find an expression for cos(pi/10). To do so, we'll use the half-angle formula for cosine:
cos(2x) = 2cos^2(x) - 1
Rearranging the formula, we get:
cos(x) = sqrt( (1 + cos(2x)) / 2 )
We can now apply the half-angle formula by letting x = pi/10:
cos(pi/10) = sqrt((1 + cos(2(pi/10))) / 2)
Remember that we know sin(pi/10) from the given information. Thus, we can substitute it in the expression for cos(pi/10):
cos(pi/10) = sqrt((1 + cos(2(pi/10))) / 2)
= sqrt((1 + cos(pi/5)) / 2)
Now, substituting both expressions of sin(pi/10) and cos(pi/10) into the previously derived equation for sin(pi/5), we get:
sin(pi/5) = 2 * [(sqrt(5)-1)/4] * [sqrt((1 + cos(pi/5)) / 2)]
Simplifying further, we have:
sin(pi/5) = (sqrt(5)-1) * sqrt((1 + cos(pi/5)) / 2) / 2
Therefore, an exact expression for sin(pi/5) using the double-angle formulas is:
sin(pi/5) = (sqrt(5)-1) * sqrt((1 + cos(pi/5)) / 2) / 2