The Cosine Rule What is actually the Cosine rule. can anyone explain it to me in way that I can understand it, explain in a simple way? that provide simple examples? (Only the Cosine rule) thanks in advance, I tried googling online but most of it is full of complexity, our school library is renovated.
 A: The cosine rule is a generalization of the Pythagorean theorem. The Pythagorean theorem says that if the legs of a right triangle have lengths $a$ and $b$, and the hypotenuse has length $c$, then $c^2=a^2+b^2$. However, this relationship holds only if the angle opposite the long side is a right angle; no other angle works. You can think of the cosine rule as giving the necessary correction when the angle isn’t a right angle. I’ll refer to the notation in the following diagram:

If angle $C$ is a right angle, we have $c^2=a^2+b^2$, by the Pythagorean theorem. You can see, though, that if you shrink angle $C$ a bit without changing the lengths of sides $\overline{AC}$ and $\overline{BC}$, side $\overline{AB}$ is going to shrink, and $c$ will be less than $a^2+b^2$. Similarly, if you increase angle $C$ a bit without changing the lengths of sides $\overline{AC}$ and $\overline{BC}$, side $\overline{AB}$ is going to stretch, and $c$ will be greater than $a^2+b^2$. The cosine rule adds in an extra term that depends on angle $C$ in order to take into account this shrinking or stretching: $$c^2=a^2+b^2-2ab\cos C\;.\tag{1}$$ The cosine of a right angle ($90$°, or $\pi/2$ radians), is $0$, so when $C$ is a right angle, $(1)$ reduces to the Pythagorean theorem: $$c^2=a^2+b^2-2ab\cdot 0 = a^2+b^2\;.$$ As angle $C$ shrinks from $90$° down towards $0$°, the cosine of the angle increases, and the $-2ab\cos C$ term subtracts more and more from $a^2+b^2$, making $c^2$ smaller and smaller, as it should be. Exactly the opposite happens as angle $C$ increases from $90$° towards $180$°.
The cosine rule is typically used in two ways. First, if you know the lengths of two sides of a triangle $-$ any triangle $-$ and the size of the angle between those two sides, you can use the cosine rule to calculate the length of the remaining side of the triangle. Call the lengths of the known sides $a$ and $b$, and let $C$ be the angle between them: then you can plug $a$, $b$, and $\cos C$ into $(1)$ to calculate $c^2$ and then $c$, thereby getting the length of the third side of the triangle. The first example here shows such a calculation.
Secondly, if you know all three sides of a triangle, you can use $(1)$ to find any of the three angles. Say that I know $a$, $b$, and $c$, and I want to find angle $C$; I just substitute the known values for $a$, $b$, and $c$ into $(1)$, and I’m left with a simple linear equation in one unknown, $\cos C$. Solving it for $\cos C$ is trivial, and one then uses a calculator (or a table of inverse cosines) to find $C$. The second and third examples on that web page are of this type.
