Question

Why isn't this triangle an ambiguous case? (where there can be 2 possible triangles from the same set of lengths and an angle)

I've figured out the height of the triangle `h=3.4202` and since that ambiguous cases happen when `h< a< b` where `a` is the length opposite of a given length and `b` is some other length, shouldn't this triangle have 2 possible answers?

I'm terribly new to the whole subject of ambiguous cases so any help would be greatly appreciated!
Thanks in advance!

Answer 1

See below.

Explanation:

This is your triangle. As you can see it is an ambiguous case.

So to find the angle `theta`:

`sin(20^@)/8=sin(theta)/10`

`sin(theta)=(10sin(20^@))/8`

`theta=arcsin((10sin(20^@))/8)=color(blue)(25.31^@)`

Because it's the ambiguous case:

Angles on a straight line add to `180^@`, so other possible angle is:

`180^@-25.31^@=color(blue)(154.69^@)`

You can see from the diagram that, as you noted:

`h< a < b`

Here is a link that may help you. This can take a while to grasp, but you seem to be on the right track.

http://www.softschools.com/math/calculus/the_ambiguous_case_of_the_law_of_sines/