The applet below illustrates all the possible extraneous solutions that could be found when working with the ambiguous case.[br][br]Drag slider AB to change the length of segment AB. [br]For some lengths of AB, there are two real solutions. [br]For other lengths - in particular, when the length of AB is "too long" and is greater than the other given side - there is one negative extraneous solution.[br]For other lengths - in particular, when the length of AB is "too short" and AB is less than [math]AC\cdot sin(C) [/math]- there are two complex extraneous solutions.
It is possible that side AB is longer than side AC. In this case, there will be two real solutions to the Law of Cosines equation, but one of the solutions will be negative, qualifying it as an extraneous solution. [br][br]If you consider the negative solution to refer to a directed line segment, then there is another triangle where the angle at C is now an exterior angle.[br][br]Drag the slider AB and drag point B to swing side AB to different locations.
In this case, the Law of Cosines equation leads to two complex conjugate solutions. What do these solutions mean geometrically?[br][br]In the 3D view, the coordinate of points O and O' give a clue as to the meaning of the complex conjugate solutions. Ignoring the y-coordinates of the points, you can deduce the complex conjugate solutions. [br][br]The geometry of the 3D view is as follows:[br][list=1][*]The circle centered at A lying in the xy-plane has a radius of AB.[/*][*]The circle centered at A lying in a plane perpendicular to the xy-plane has a radius of [math]AC\cdot sin(C)[/math] - or the length of AB if triangle ABC were a right triangle with B the right angle.[/*][*]The complex conjugate solutions are [math]AC^2-\left(AC\cdot sin\left(C\right)\right)^2\pm i\cdot\left(\left(AC\cdot sin(C)\right)^2-BC^2\right)[/math][/*][/list]