Is it possible to have a triangle such that an unknown side can't be determined using trigonometric ratios, the Sine Law, or the Cosine Law? If so, provide an example.

Question

Is it possible to have a triangle such that an unknown side can't be determined using trigonometric ratios, the Sine Law, or the Cosine Law? If so, provide an example.

Answer 1

I agree with Paul, it is the SSA case that is ambiguous (may be able to find two solutions for missing side, or find one solution or find there is no side possible given the constraints).

how about 3,3,90 degrees. I think no possible solution. I disagree with premise of question because it is the law of cosines used to determine the resolution of the ambiguity.

in this case (cos90=0), and calling 90 degree angle B so sides given are b and c

3²=3²+a²-2*3*3*0

there is no side a>0

Answer 2

In order to solve a triangle you must have the same parts as you would to determine that 2 triangles are congruent in plane geometry (SAS, ASA, SSS, AAS) and if you have those parts, the triangle is determined uniquely. If you have an angle and 2 sides which do not include the angle, then there are 4 possibilities which you should be able to list if you think about it.