Trigonometry/Law of Cosines

Law of Cosines


The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines:
 * $$a^2+b^2-2ab\cos(\theta)=c^2$$

where $$\theta$$ is the angle between sides $$a$$ and $$b$$.

Does the formula make sense?
This formula had better agree with the Pythagorean Theorem when $$\theta=90^\circ$$.

So try it...

When $$\theta=90^\circ$$, $$\cos(\theta)=\cos(90^\circ)=0$$

The $$-2ab\cos(\theta)=0$$ and the formula reduces to the usual Pythagorean theorem.

Permutations
For any triangle with angles $$A,B,C$$ and corresponding opposite side lengths $$a,b,c$$, the Law of Cosines states that
 * $$a^2=b^2+c^2-2bc\cdot\cos(A)$$
 * $$b^2=a^2+c^2-2ac\cdot\cos(B)$$
 * $$c^2=a^2+b^2-2ab\cdot\cos(C)$$

Proof


Dropping a perpendicular $$OC$$ from vertex $$C$$ to intersect $$AB$$ (or $$AB$$ extended) at $$O$$ splits this triangle into two right-angled triangles $$AOC$$ and $$BOC$$, with altitude $$h$$ from side $$c$$.

First we will find the lengths of the other two sides of triangle $$AOC$$ in terms of known quantities, using triangle $$BOC$$.


 * $$h=a\sin(B)$$

Side $$c$$ is split into two segments, with total length $$c$$.
 * $$\overline{OB}$$ has length $$\overline{BC}\cos(B)=a\cos(B)$$
 * $$\overline{AO}=\overline{AB}-\overline{OB}$$ has length $$c-a\cos(B)$$

Now we can use the Pythagorean Theorem to find $$b$$, since $$b^2=\overline{AO}^2+h^2$$.


 * $$b^2$$
 * $$=\bigl(c-a\cos(B)\bigr)^2+a^2\sin^2(B)$$
 * $$=c^2-2ac\cos(B)+a^2\cos^2(B)+a^2\sin^2(B)$$
 * $$=a^2+c^2-2ac\cos(B)$$
 * }
 * $$=a^2+c^2-2ac\cos(B)$$
 * }
 * $$=a^2+c^2-2ac\cos(B)$$
 * }

The corresponding expressions for $$a$$ and $$c$$ can be proved similarly.

The formula can be rearranged:
 * $$\cos(C)=\frac{a^2+b^2-c^2}{2ab}$$

and similarly for $$cos(A)$$ and $$cos(B)$$.

Applications
This formula can be used to find the third side of a triangle if the other two sides and the angle between them are known. The rearranged formula can be used to find the angles of a triangle if all three sides are known. See Solving Triangles Given SAS.