Just wanted to share this, I worked it out a couple of years ago and was surprised by the accuracy:
Near Exact Trisection: 1. Start with an unknown angle <90 deg., label the vertex A. 2. Draw an arc with origin at A crossing both lines of the angle at points B and C. 3. Draw line BC making an isosceles triangle. 4. Using point C as the origin, draw an arc crossing line BC and the earlier arc somewhere between ¼ and ½ way between points C and B. Label where this new arc crosses line BC point D. Label where this new arc crosses the first arc point E. 5. Draw line DE and extend it well past A . If line DE passes exactly through point A (it wont) stop, your first guess was an exact trisection. 6. Extend line AC well past point A, step off 3 times length AC from point A and label the new point F. 7. Swing an arc of length AF with A as the origin that crosses the extended line DE near point F. Label the intersection G. 8. Draw line GA and extend it to intersect the original arc from step 2. Label the intersection E.
Line AE is a good (within less than 1/1000 degree) trisection. However this is only the start. Repeating the process from step 4 using CE as the arc radius results in a trisection to within 10E-11 degrees. Each subsequent iteration improves the trisection by several orders of magnitude.