0 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 .
+129830 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
