Question

How do you figure out if there is a triangle, or no triangle?

I'm getting vex...

For example:

a=10, b=16.6, A=27 degree (produces two triangles).

Example 2:

a=18, b=25, A=70 degree (No Triangle).

Please explain, why.

Answer 1

I did notice that these two problems are quiet different.

Answer 2

I'm thinking that one way to see it is with the law of sines...

Answer 3

\[{a \over \sin A}={b \over \sin B}\to \sin B={b \over a} \sin A={25\over18}\sin(70)\approx1.3\]but trig functions can never be greater than one, so \[B=\sin^{-1}(1.3)\]is undefined, therefor no triangle.

Answer 4

for the first triangle \[\sin B={16.6\over10}\sin(27)\approx0.75<1\]so angle B can be calculated, hence this obeys the law of sines and is a triangle.

Answer 5

Thank you much.