# 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?

Aug 5, 2015

See explanation.

#### Explanation:

Find $\sin B$

$\sin B = \frac{12 \sin \left({35}^{\circ}\right)}{9.1} \approx 0.756365$

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

${\sin}^{-} 1 \left(0.756365\right) \approx {49.14}^{\circ}$

So $B = {49.14}^{\circ}$ is one solution.
(And $C = 180 - {\left(35 + 49.14\right)}^{\circ} = {95.86}^{\circ}$)

Another angle with the same sine is ${180}^{\circ} - {49.14}^{\circ} = {130.86}^{\circ}$

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