Solving all possible triangles?

Question

So we're doing oblique triangles -- Law of Sines and all that good stuff =).

I have a bunch of problems that ask you to solve for "all possible triangles that satisfy the given conditions".

For example, one gives $b=45, c=42, \angle C = 38^ \circ$.

This is using the convention that $a$ is opposite of $\angle A$, $b$ is opposite of $\angle B$, and $c$ is opposite of $\angle C$.

How do I go about solving "all possible triangles"?

Thanks!

evamvid

Answer 1

Hope the following sketch helps:- .

Answer 2

Using Law of Sines we have:

$$\frac b{\sin B} = \frac c{\sin C} \iff \sin B = \frac{b \cdot \sin C}{c}$$

Now solving this you'll get $\angle B \approx 41,27^{\circ}$ or $\angle B \approx 138.72^{\circ}$

Now check both cases. Use $A+B+C = 180^{\circ}$ and Law of Sines to get 2 solutions.