====== Proof of the Law of Cosines ======

{{:en:law-of-cosines1-fs8.png?nolink&}}

===== Statement =====

For a triangle with sides $a$, $b$, and $c$, and angle $\theta$ between sides $b$ and $c$, the **Law of Cosines** states:

$$
a^2 = b^2 + c^2 - 2bc\cos(\theta)
$$

-----

===== Proof =====

We construct the proof by splitting the triangle into two right-angled triangles.

Drop a perpendicular from the vertex opposite side $a$ onto side $c$.

{{:en:law-of-cosines2-fs8.png?nolink|}}

==== Step 1: Identify the lengths ====

Using trigonometry in the smaller right triangle:

  * The vertical height (altitude) is:
    $$
    b \sin(\theta)
    $$

  * The horizontal projection of side $b$ onto side $c$ is:
    $$
    b \cos(\theta)
    $$

  * Therefore, the remaining horizontal segment is:
    $$
    c - b \cos(\theta)
    $$

{{:en:law-of-cosines3-fs8.png?nolink|}}

==== Step 2: Apply the Pythagorean Theorem ====

Consider the right triangle formed with:

  * Vertical side: $b \sin(\theta)$
  * Horizontal side: $c - b \cos(\theta)$
  * Hypotenuse: $a$

By the Pythagorean Theorem:

$$
a^2 = (b \sin(\theta))^2 + (c - b \cos(\theta))^2
$$

==== Step 3: Expand the expression ====

$$
\begin{aligned}
a^2 &= b^2 \sin^2(\theta) + (c - b \cos(\theta))^2 \\
\\
&= b^2 \sin^2(\theta) + c^2 - 2bc\cos(\theta) + b^2 \cos^2(\theta)
\end{aligned}
$$

Rearrange terms:

$$
a^2 = b^2 \sin^2(\theta) + b^2 \cos^2(\theta) + c^2 - 2bc\cos(\theta)
$$

==== Step 4: Use a trigonometric identity ====

Recall the identity:

$$
\sin^2(\theta) + \cos^2(\theta) = 1
$$

Substitute:

$$
a^2 = b^2(1) + c^2 - 2bc\cos(\theta)
$$

-----

===== Final Result =====

$$
a^2 = b^2 + c^2 - 2bc\cos(\theta)
$$

-----

===== Why this works (Intuition) =====

The proof works by:

  * Breaking the triangle into right triangles (where we can use the Pythagorean Theorem)
  * Expressing unknown lengths using sine and cosine
  * Reassembling everything into a single equation

The extra term $-2bc\cos(\theta)$ accounts for the fact that the triangle is not necessarily a right triangle.

-----

===== Summary =====

  * The Law of Cosines generalizes the Pythagorean Theorem
  * It works for **all triangles**, not just right-angled ones
  * It is essential in geometry, physics, engineering, and computer graphics