Quick questions on The cosine rule for HSC Maths Standard 2

For any triangle $ABC$ with sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$:

Rearranging the same equation to make the angle the subject:

The included angle is the one between the two given sides. In worded problems it shows up as "the angle at $B$ between $AB$ and $BC$", "an angle of $60\degree$ between the two roads", or as the bend in a journey. If the angle you are given is not enclosed by your two sides, you do not have a clean SAS setup, so re-read or redraw before substituting.

For an obtuse included angle, $\cos C$ is negative, so $-2ab\cos C$ becomes a positive number you add. Forgetting this is a frequent error. For example with $C = 110\degree$, $\cos 110\degree \approx -0.342$, and $-2ab\cos C = +0.684 ab$. The opposite side comes out longer than either enclosing side, which matches the geometry of a wide-open triangle.

$\cos$ of an angle between $90\degree$ and $180\degree$ is negative, so $-2ab\cos C$ adds to the total.