The cosine rule (AQA GCSE Further Maths): Revision Notes

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The cosine rule

What is the cosine rule?

The cosine rule is a powerful mathematical tool that helps us find unknown sides or angles in any triangle. Unlike the sine rule, the cosine rule is particularly useful when you have specific information about a triangle - either when you know two sides and the included angle, or when you know all three sides but need to find an angle.

The cosine rule works for any triangle, whether it's acute, obtuse, or right-angled, making it more versatile than some other triangle-solving methods.

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The cosine rule is especially powerful because it works where the sine rule might fail. When you don't have a complete side-angle pair, or when you're dealing with the ambiguous case of the sine rule, the cosine rule provides a reliable alternative.

The two main formulas

There are two versions of the cosine rule, and choosing the right one depends on what you're trying to find:

For finding a side length (when you know two sides and the included angle): a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc \cos A

For finding an angle (when you know all three sides): cos⁡A=b2+c2−a22bc\cos A = \frac{b^2 + c^2 - a^2}{2bc}

In these formulas, the letters aa, bb, and cc represent the lengths of the sides of the triangle, while AA, BB, and CC represent the angles opposite to those sides respectively. This is standard triangle notation that you'll see throughout geometry.

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Choosing the Right Formula:

  • Use a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc \cos A when you have SAS (Side-Angle-Side) information
  • Use cos⁡A=b2+c2−a22bc\cos A = \frac{b^2 + c^2 - a^2}{2bc} when you have SSS (Side-Side-Side) information

Remember: "SAS for sides, SSS for angles"

The cosine rule can also be written in alternative forms depending on which side or angle you're looking for:

  • b2=a2+c2−2accos⁡Bb^2 = a^2 + c^2 - 2ac \cos B
  • c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab \cos C

How the cosine rule works - the proof

Understanding where the cosine rule comes from helps you remember and apply it correctly. The rule is derived using the Pythagorean theorem applied to a cleverly constructed diagram.

Figure

The proof starts with triangle ABC, where we draw a perpendicular line (altitude) from vertex C down to the base AB, creating point D. This altitude has height hh and divides the base into two segments of lengths xx and (c−x)(c-x).

This creates two right-angled triangles: triangle ACD and triangle BCD. By applying the Pythagorean theorem to both triangles and using trigonometric relationships, we can derive the cosine rule formula through algebraic manipulation.

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Key Insight in the Proof: The crucial step is recognising that cos⁡A=xb\cos A = \frac{x}{b}, which means x=bcos⁡Ax = b \cos A. When we substitute this into our Pythagorean relationships and simplify, we arrive at the familiar cosine rule formula: a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc \cos A.

Think of the cosine rule as "Pythagoras with an extra correction term" - the −2bccos⁡A-2bc \cos A part adjusts for the angle that isn't 90°.

Worked example - finding a side length

Let's see how to use the cosine rule to find an unknown side when you know two sides and the included angle.

Figure

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Worked Example: Finding a Side Length

Problem: In triangle ABC, AC = 8 cm, BC = 9 cm, and angle C = 37°. Find the length of side AB.

Solution: Since we know two sides and the included angle, we use the first form of the cosine rule: c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab \cos C

Substituting our known values: c2=92+82−2×9×8×cos⁡37°c^2 = 9^2 + 8^2 - 2 \times 9 \times 8 \times \cos 37° c2=81+64−144×cos⁡37°c^2 = 81 + 64 - 144 \times \cos 37° c2=145−144×0.7986c^2 = 145 - 144 \times 0.7986 c2=29.996c^2 = 29.996

Therefore: AB=29.996=5.48AB = \sqrt{29.996} = 5.48 cm (to 3 significant figures)

Worked example - finding an angle

Now let's see how to find an unknown angle when you know all three sides.

Table

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Worked Example: Finding an Angle

Problem: In triangle PQR, PQ = 8.2 cm, PR = 5.3 cm, and QR = 12.1 cm. Find angle P.

Solution: Since we know all three sides, we use the second form of the cosine rule: cos⁡P=q2+r2−p22qr\cos P = \frac{q^2 + r^2 - p^2}{2qr}

Where qq and rr are the sides adjacent to angle P, and pp is the opposite side. cos⁡P=5.32+8.22−12.122×5.3×8.2\cos P = \frac{5.3^2 + 8.2^2 - 12.1^2}{2 \times 5.3 \times 8.2} cos⁡P=28.09+67.24−146.4186.92\cos P = \frac{28.09 + 67.24 - 146.41}{86.92} cos⁡P=−51.0886.92\cos P = \frac{-51.08}{86.92} cos⁡P=−0.588\cos P = -0.588

Therefore: P=cos⁡−1(−0.588)=126.0°P = \cos^{-1}(-0.588) = 126.0° (to 1 decimal place)

Note that this gives us an obtuse angle, which makes sense since the longest side (12.1 cm) is opposite to this angle.

Common mistakes and exam tips

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Common Mistakes to Avoid:

Calculator errors: When working with the cosine rule on non-calculator papers, evaluate the terms a2a^2, b2b^2, and 2abcos⁡C2ab \cos C separately. A common mistake is to calculate a2+b2−2aba^2 + b^2 - 2ab first and then multiply by cos⁡C\cos C, but this is incorrect. The correct order of operations is crucial.

Forgetting to square root: When finding a side length, remember that the cosine rule gives you the square of the side length. Don't forget to take the square root to get your final answer!

Using the wrong formula: Make sure you choose the right version of the cosine rule. Use a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc \cos A when finding a side, and cos⁡A=b2+c2−a22bc\cos A = \frac{b^2 + c^2 - a^2}{2bc} when finding an angle.

Obtuse angles: When finding angles, don't be surprised if you get an obtuse angle (greater than 90°). The cosine of an obtuse angle is negative, which is perfectly normal in triangle calculations.

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Key Points to Remember:

  • The cosine rule works for ANY triangle and is especially useful when you know two sides and the included angle (SAS) or all three sides (SSS)

  • There are two main forms: a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc \cos A for finding sides, and cos⁡A=b2+c2−a22bc\cos A = \frac{b^2 + c^2 - a^2}{2bc} for finding angles

  • The cosine rule is derived from the Pythagorean theorem and can be thought of as "Pythagoras with an extra correction term"

  • Always check your calculator work carefully and remember to square root when finding side lengths

  • Be prepared for obtuse angles when finding angles - a negative cosine value indicates an angle greater than 90°

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