ABCD is a square where M and N are midpoints of AD and CD, respectively. If sin∠MBN=a/b, where a and b are coprime positive integers, what is the value of a+b?

Answer 1

If we let x = ∠MBN and y=∠ABM=∠NBC, then
x+2y = pi/2
sin x = sin(pi/2-2y) = cos 2y = 1-2sin^2 y
Now, siny = 1/√5, so
sinx = 1 - 2(1/5) = 3/5
a+b=8