+1439 In square ABCD, P is on BC such that BP = 4 and PC = 1, and Q is on line CD such that DQ = 1 and QC = 4. Find sin angle PAQ.
+129850 A B
4
P
1
D 1 Q 4 C
P = (5 , 1)
Q = (1,0)
A = (0, 5)
PA = sqrt 41
PQ = sqrt 17
AQ = sqrt 26
Law of Cosines
PQ^2 = AQ^2 + PA^2 - 2(AQ * PA) * cos (PAQ)
17 = 26 + 41 - 2 ( sqrt 26 * sqrt 41) cos (PAQ)
cos (PAQ) = [ 17 - 26 - 41 ] / [ - 2 sqrt 1066 ] = -50 / [ -2 sqrt 1066 ] = 25 / sqrt (1066)
sin (PAQ) = sqrt [ 1 - (25 / sqrt 1066)^2 ] = sqrt [ 1 - 625 / 1066 ] = sqrt [ 441 / 1066 ] =
21 / sqrt 1066 = (21 sqrt (1066) / 1066
