Answer 6
If we restrict the discussion to unit vectors at various angles (A), the X component is cos(A) and the Y component is sin(A), and the correct magnitude is 1. Furthermore, let's concentrate on the angles between 0 and 45°. You know that both cos(A) and sin(A) are positive and that cos(A) > sin(A). (The absolute value and compare operations provide the symmetry that covers the rest of the unit circle.) The approximation then gives the result:

Graphing this shows that this is most negative (0.943) at 45° and most positive (1.054) at approximately 18.4° (the actual angle is given by tan(1/3) ). The peak error is therefore –5.7%, +5.4%.

Contributor: Dave Tweed

Published April 2003