Foundations — Law of cosines — proof and applications

Before you can even read the sentence $c^2=a^2+b^2-2ab\cos C$, you must own every mark in it: the letters, the little $2$ that means "squared", the $\cos$, and the idea of an angle facing a side. This page assumes nothing and hands you each piece with its picture.

0. What a triangle's parts are called

Every symbol in this topic hangs off one labelled triangle, so we draw it first.

The picture to burn in: stand at corner $A$ and look across the empty middle of the triangle — the side you are staring at is $a$. That "looking across" is what opposite means. Every rule in this chapter reuses this pairing, so if you mix up which side faces which angle, every formula breaks.

1. The little raised $2$ — "squared"

The formula is full of $a^2$, $b^2$, $c^2$. That raised $2$ is the whole reason the formula feels heavy, so let us make it concrete.

The picture: $a^2$ is literally the area of a square whose side has length $a$. A side of length $3$ builds a square of area $3\times 3=9$.

2. Angle, and the special mark $90°$

The picture to keep: think of the two sides meeting at corner $C$ as two sticks joined by a hinge. The angle $C$ is how far open the hinge is.

• Hinge barely open → small angle (called acute, less than $90°$).
• Hinge at a perfect corner → exactly $90°$ (called a right angle, marked with a tiny square).
• Hinge opened past the corner → wide angle (called obtuse, between $90°$ and $180°$).

3. The included angle (the hinge between two sides)

Picture: sides $a$ and $b$ both touch corner $C$. The angle wedged between them, right at $C$, is their included angle. Notice it is exactly $C$ — and $C$ faces side $c$ (from §0). This is the golden triangle of the whole topic:

Get this and the formula $c^2=a^2+b^2-2ab\cos C$ reads like a sentence: "the gap-squared depends on the two arm-lengths squared, minus a correction that depends on how wide you opened the hinge."

4. $\cos$ — the "how wide is the hinge?" dial

Here is the one genuinely new tool. Why do we need it at all? Because "how open the hinge is" (the angle) and "how far apart the far ends are" (a length) are different kinds of thing — one is a turn, one is a distance. We need a translator that converts an angle into a plain number we can multiply into a length formula. That translator is the cosine.

The picture — the unit circle: draw a circle of radius $1$. Sweep an arm out at angle $C$ from the rightward direction. The arm's horizontal reach (how far right or left its tip landed) is $\cos C$.

5. $\arccos$ — running the dial backwards

Why the topic needs it: in the SSS case you know the three sides, so you can compute the number $\cos C$ — but you actually want the angle. $\arccos$ is the "undo" button that turns the cosine-number back into degrees. $\cos$ goes angle $\to$ number; $\arccos$ goes number $\to$ angle. They cancel each other.

6. Coordinates $(x,y)$ — the address of a point

The parent's proof drops the triangle onto a grid, so you need one last piece of language.

Picture: the origin is home base. $(b,0)$ means "walk $b$ steps right, $0$ steps up" — you stay on the ground line. $(a\cos C,a\sin C)$ is the tip of the other arm, using cosine for its rightward reach (§4) and $\sin$ (sine, the upward reach — cosine's vertical twin) for its height. This is precisely why the proof can write $B=(a\cos C,a\sin C)$: the arm of length $a$ swung to angle $C$ lands there.

How these foundations feed the topic

Read top to bottom: the plain triangle language and squaring feed Pythagoras; angles and the hinge feed cosine; cosine plus Pythagoras plus coordinates assemble the Law of Cosines; and arccosine lets you run it backwards for the SSS case.

Equipment checklist

Test yourself — cover the right side and answer aloud.

Connections

• Pythagoras Theorem — the squaring-and-areas idea these foundations lead into.
• Unit Circle and Cosine Values — the full picture behind $\cos$ going negative.
• Dot Product — coordinates + cosine reappear here as the same fact.
• Solving Triangles — where the SAS/SSS vocabulary gets used.
• Law of Sines — the partner rule you choose between.
• Parent: Hinglish version →