That might work, although it will have the same branch cut issue as the parametric solution. Recalling vaguely from geometry, SAS (side-angle-side) SSS and ASA all uniquely specify a triangle, but SSA does not. The figure you drew has this SSA arrangement and your triangle is not unique. I can make a second triangle with the same SSA - by picking the same other point, behind the car.

Not to say you couldn't tackle the problem going that way - always more than one way to get the job done. Wikipedia's article on the law of sines appears to have a special section on this, entitled "the ambiguous case".