The Cosine Rule: A Bridge Between Sides and Angles
From Pythagoras to a Universal Rule
We begin our journey with a familiar friend: the Pythagorean Theorem. For a right-angled triangle with hypotenuse $c$, it states $a^2 + b^2 = c^2$. This works perfectly, but only when one angle is exactly 90°. What happens if the angle is acute or obtuse? The relationship between the sides changes. The cosine rule generalizes Pythagoras' idea for any triangle.
Consider a triangle with vertices $A$, $B$, and $C$. The sides opposite these angles are labeled $a$, $b$, and $c$, respectively. The standard form of the cosine rule is:
To find a side: $a^2 = b^2 + c^2 - 2bc \cos A$
To find an angle: $\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}$
Notice the symmetry. You can rotate the letters: $b^2 = a^2 + c^2 - 2ac \cos B$ and $c^2 = a^2 + b^2 - 2ab \cos C$ are equally valid. The crucial part is that the angle in the formula must be the angle opposite the side you are calculating or finding.
Applying the Rule: Finding Missing Sides and Angles
Let's break down the two main use cases with clear, step-by-step examples.
Case 1: Finding a Side (SAS[1])
You know two sides and the angle between them. For instance, in $\triangle PQR$, side $p = 7$ cm, side $q = 10$ cm, and the angle between them, $R = 60°$. We want side $r$, which is opposite angle $R$.
- Write the formula for side $r$: $r^2 = p^2 + q^2 - 2pq \cos R$.
- Substitute the known values: $r^2 = 7^2 + 10^2 - 2(7)(10) \cos 60°$.
- Calculate: $r^2 = 49 + 100 - 140 \times 0.5$.
- Simplify: $r^2 = 149 - 70 = 79$.
- Final step: $r = \sqrt{79} \approx 8.89$ cm.
Case 2: Finding an Angle (SSS[2])
You know all three sides. In $\triangle XYZ$, $x = 8$ m, $y = 5$ m, $z = 7$ m. Find angle $Y$ (opposite side $y$).
- Write the formula for $\cos Y$: $\cos Y = \dfrac{x^2 + z^2 - y^2}{2xz}$.
- Substitute: $\cos Y = \dfrac{8^2 + 7^2 - 5^2}{2 \times 8 \times 7}$.
- Calculate: $\cos Y = \dfrac{64 + 49 - 25}{112} = \dfrac{88}{112} = 0.7857$.
- Use the inverse cosine function: $Y = \cos^{-1}(0.7857) \approx 38.2°$.
Side-by-Side: Cosine Rule vs. Sine Rule
Another important triangle-solving tool is the sine rule. Knowing when to use which rule is key to efficient problem-solving. The table below clarifies the differences.
| Aspect | Cosine Rule | Sine Rule |
|---|---|---|
| Data Required | SAS or SSS. | AAS[3] or ASA[4] (two angles and any side). |
| Best Used For | Finding a third side given two sides and the included angle; or finding an angle given all three sides. | Finding an unknown side or angle when you have a known angle-side opposite pair. |
| Formula (example) | $a^2 = b^2 + c^2 - 2bc \cos A$ | $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$ |
| Ambiguous Case | No ambiguous case. A unique solution always exists for valid inputs. | Yes, when given SSA[5] (two sides and a non-included angle), zero, one, or two triangles are possible. |
Navigating Real-World Terrain with the Cosine Rule
The cosine rule isn't just for textbooks; it's a practical tool for measurement and calculation where direct measurement is impossible.
Example 1: Surveying a Plot of Land
A surveyor wants to find the distance across a swamp ($AC$). She marks points $A$, $B$, and $C$ on dry land, forming a triangle. She measures $AB = 120$ m, $BC = 100$ m, and the angle at $B$ between these sides is $70°$. To find $AC$, she uses the cosine rule: $AC^2 = 120^2 + 100^2 - 2(120)(100)\cos 70°$. Calculating gives $AC \approx 126.5$ m, allowing her to map the area without crossing the swamp.
Example 2: Analyzing Forces in Physics
Two forces act on an object: a 10 N force and a 15 N force, with an angle of 50° between them. The magnitude of the resultant force is found using the cosine rule, treating the force vectors as sides of a triangle: $R^2 = 10^2 + 15^2 - 2(10)(15)\cos(130°)$. Note the angle is $130°$ because the angle between the forces when placed tail-to-tail is the supplement of 50°. This yields $R \approx 22.1$ N.
Important Questions
A: The most common derivation uses the Pythagorean theorem. For a triangle $ABC$, drop a perpendicular from $C$ to side $c$, creating two right triangles. Using Pythagoras on each and some algebraic manipulation (involving the definition of cosine: $\cos A = \text{adjacent}/\text{hypotenuse}$) leads directly to the formula $a^2 = b^2 + c^2 - 2bc \cos A$.
A: Absolutely, and it simplifies beautifully. If angle $A = 90°$, then $\cos 90° = 0$. The cosine rule $a^2 = b^2 + c^2 - 2bc(0)$ becomes $a^2 = b^2 + c^2$, which is the classic Pythagorean theorem. Thus, the cosine rule is a true generalization.
A: A useful mnemonic is: "The square of one side equals the sum of squares of the other two sides, minus twice their product times the cosine of the included angle." For the angle formula, remember it's a rearrangement: "Cosine of angle equals (sum of squares of adjacent sides minus square of opposite side) divided by (twice the product of adjacent sides)."
The cosine rule is an indispensable and versatile tool in geometry. It elegantly bridges the gap between the sides and angles of any triangle, right-angled or not. By mastering its two forms—for finding a missing side (SAS) and for finding a missing angle (SSS)—you unlock the ability to solve a vast array of geometric and real-world problems. Its relationship to the Pythagorean theorem provides a satisfying link between a special case and a universal law. From mapping land to resolving vectors, the cosine rule proves that a simple formula can have powerful and wide-reaching applications.
Footnote
[1] SAS: Side-Angle-Side. A configuration where two sides and the angle between them are known.
[2] SSS: Side-Side-Side. A configuration where the lengths of all three sides are known.
[3] AAS: Angle-Angle-Side. Two angles and a side not between them are known.
[4] ASA: Angle-Side-Angle. Two angles and the side between them are known.
[5] SSA: Side-Side-Angle. Two sides and an angle not between them are known. This is the ambiguous case for the sine rule.