Hello, ceroseven!

2.In $\displaystyle \Delta ABC\!:\;\;B=65^o,\;\;b=2,\;\;c=12$

How many triangles are possible? Did you make a sketch?

If such a triangle were possible, it might look like this:

Code:

C
*
/ \
/ \
/ \
/ \ 2
/ \
/ \
/ 65° \
B * - - - - - - - * A
12

Law of Sines: . $\displaystyle \frac{\sin C}{c} \:=\:\frac{\sin B}{b}$

So we have: .$\displaystyle \frac{\sin C}{12} \:=\:\frac{\sin 65^o}{2} \quad\Rightarrow\quad \sin C \:=\:\frac{12\sin65^o}{2} \:\approx\:5.44$

Then: .$\displaystyle C \:=\:\arcsin(5.44) \:=\:??? $ . . . which does not exist.

Therefore, it is *no* triangle.