The graph appears to show three real solutions:

The only question might be Does it "wiggle around" near x = 0"? But x^3 and sinx both become negative on the negative-x side and positive on the positive-x side, and x^2 and cos x are also even functions.  The function goes down both to the left and the right of the y-axis and is an even function.

So there are indeed three distinct real roots.