Question

How do you determine the number of possible solutions, if any, using the rule for law of sines(ambiguous case) given A=35*, b=12, a=9.1?

Answer 1

See explanation.

Explanation:

Find `sinB`

`sinB = (12sin(35^@))/9.1~~ 0.756365`

OK, that is less that or equal to `1`, so there is a solution.

`sin^-1(0.756365) ~~ 49.14^@`

So `B = 49.14^@` is one solution.
(And `C = 180 -(35+49.14)^@ = 95.86^@`)

Another angle with the same sine is `180^@ - 49.14^@ = 130.86^@`

Add the `35^@` angle we started with, to get: `165.86`.
We have not gone over the `180^@` total for the angles of a triangle, so `B = 130.86^@` will give us a second solution.
(And `C = 180 -(35+130.86)^@ = 14.14^@`)