I'm not entirely sure how you actually find how far away the man is from the cave entrance at that point, but I'd guess it's about 70m.

Anyway, here's what I did to set up the problem. The stuff in red is what we're given and the stuff in blue and the 55 and 71 are as far as I got in solving.

See how you can make a 30-60-90 triangle from this? And remember that the ratios for the sides of a 30-60-90 triangle are 1:√3:2 (short side, long side, hypotenuse). The long side is 100. Divided by the √3 is about 71, which is the length of the short side. The hypotenuse is double that, so 142. So yeah... 87 is close to half the hypotenuse, but it's a little more than that, really.

Unfortunately, you don't form a right triangle if you connect the rest of the hypotenuse and short side of the triangle (whose lengths we know) and the line that would be formed if the man walked straight towards the cave entrance from where he is right now. It kinda looks like a right triangle in my picture above but it's not. It's been a LONG time since I had trig, but I don't THINK you can use the law of sines or something else to get the exact distance the man is to the cave entrance at this point unless you had another piece of information. If someone has an idea for how to solve it beyond that, they can take it from there.

Hope that helps.