Math Is Fun Forum

Using Bob Bundy's diagram and measuring the distance below AB by tracing through the points ALKMB
we get AL -LE +FM -MD =0
or   4 -3 cos x    + 4 sin x   -5 cos x = 0

or sin x - 2 cos x = - 1

between 0° and 90°  both sin x and -2 cos x are increasing so only one single solution.
And by enlightened guesswork (and phrontister's hint of 345 triangles) we find that

sin x = 3/5  =>  cos x = 4/5   and this solves sin x -2 cos x =35/ - 2*4/5 = -5/5 = -1

This is a lot simpler than Bob Bundy's method of converting sin s - 2 cos x  into K sin (x +α)
and reduces it from what I guess is Further Maths A-level to GCSE

So Yes it is using 345 triangles

And by a similar route to the above
AB = EK +K F + DB

  = 3 sin x  + 4 cos x  + 5 sin x
  = 8 sin x + 4 cos x
  = 8 x 3/5 + 4 * 4/5
  = (24 + 16)/5

AB  = 8