Square

Let's first find the side lengths of PAQ with the given information:

The square's side length is 4 units

Side AQ= 1^2+4^2= 17 so    AQ= sqrt17

Side PQ= 2^2+3^2=13 so    PQ= sqrt13

Side AP= 2^2+4^2=20 so     AP= 2sqrt5

Now let's use SOHCAHTOA:

We're going to do the opposite over hypotenuse, depending on which angle you are using, that should give you your answer .

Using the Law of Cosines

PQ^2  = PA^2 + AQ^2  - 2(PA*AQ) cos (PAQ)

13  =  20  + 17  - 2 (sqrt (20) * sqrt (17))cos(PAQ)

[13 - 20 - 17 ] / [ -2 sqrt (340) ]  =  cos (PAQ)  = 6/sqrt (85)

sin PAQ  = sqrt [ 1  - (6/sqrt85)^2 ] / sqrt (85) = sqrt [ 1 -36/85 ] / sqrt (85)  = sqrt [49 / 85 ] / sqrt [85] =

7 / 85