Given $\displaystyle \triangle ABC$, where $\displaystyle b=3$, $\displaystyle c=3$, $\displaystyle B=45^\circ$, and $\displaystyle C=90^\circ$, what's the length of a?

The triangle doesn't exist, because the hypotenuse cannot be the same length as one of the other sides. But when you put it in the first cosine theorem, it gives an answer

$\displaystyle a=3\cos 90^\circ + 3\cos 45^\circ=\frac{3\sqrt{2}}{2}$

???