Let point A be at (1,1). Let point B be at (2,3) and point C at (4,0).
Let theta = angle BAC. Then we can write cos theta = frac{x}{sqrt{2}} for some value of x. What is x?
Distance from A to C = sqrt [ (1- 4)^2 + (1 - 0)^2 ] = sqrt [ (-3)^2 + (1)^2 ] = sqrt (10)
Distance from A to B = sqrt [ (1 - 2)^2 + (1 - 3)^2 ] = sqrt [ (-1)^2 + (-2)^2 ] = sqrt (5)
Distance fromB to C = sqrt [ (4 - 2)^2 + (3 - 0)^2 ] = sqrt [ 2^2 + 3^2 ] = sqrt [13]
So........using the Law of Cosines we have that
[sqrt (13)]^2 - [sqrt(10)]^2 - [sqrt (5)]^2
________________________________ = cos theta
- 2 sqrt (10)*sqrt(5)
[ 13 - 10 - 5 ]
__________ = cos theta
-2 sqrt (50)
[-2]
________ = cos theta
-2 sqrt (50)
1
_______ = cos theta
sqrt (50)
1
_______ = cos theta
5 sqrt (2)
So x = 1/5