Draw this answer on x-axis and y-axis for better understanding.

Let your tent is at a point, we say it O (0,0), Joe's tent is at Point J (in the first quadrant) and Karl's tent is at Point K (in the fourth quadrant). So now acc. to the ques. line joining OJ will make angle of 37.0 and line joining OK will make angle of 23.0 with the x-axis.

Given, OJ = 21m. and OK = 32m.

Therefore; vector OJ = (21xcos23)i - (21xsin23)j

and vector OK = (32xcos37)i + (32xsin37)j

From vector triangle,

OJ + JK = OK => JK = OK - OJ

vector JK = (32xcos37 - 21xcos23)i + (32xsin37 + 21xsin23)j

|JK|^2 = [(32xcos37 - 21xcos23)^2 + (32xsin37 + 21xsin23)^2] m^2

Jk         = 28.16m

So, the distance between Joe and Karl tent is = 28.16m or 28.20m