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

enter image source here

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!

1 Answer
Apr 9, 2018

See below.

Explanation:

enter image source here

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/