Cosine Rule

Video lesson3 worked examples

Theory

The Cosine Rule is used to find a missing side when two sides and the included angle are known, or to find a missing angle when all three sides are known.

Both forms of the formula are provided on the exam sheet:

a^2 = b^2 + c^2 - 2bc\cos A

\cos A = \frac{b^2 + c^2 - a^2}{2bc}

Selecting the Right Rule

With three tools available, the marks often go to choosing correctly. Look at what you are given:

Two sides + the angle between them → area. Use $A = \tfrac{1}{2}ab\sin C$ if the question asks for the area.
Two sides + the angle between them → third side. Use the Cosine Rule.
All three sides → any angle. Use the Cosine Rule (rearranged form).
A matching side-and-angle pair → anything else. Use the Sine Rule.

The quickest test: if you can see a side and the angle opposite it, the Sine Rule will work. If you cannot, it must be the Cosine Rule.

The Golden Rule: reach for the Cosine Rule when the Sine Rule cannot start — that is, when you have two sides and the angle between them (to find the third side), or all three sides (to find any angle). The angle in the first form is always opposite the side you are finding.

⚠️ Common Examiner Traps

Order of operations: work out $2bc\cos A$ as one quantity, then subtract — don't subtract before multiplying.
Square-rooting: the formula gives $a^2$ ; remember the final square root to get $a$ .
Negative cosine = obtuse: if $\cos A$ comes out negative, the angle is obtuse (over 90°) — the calculator handles this automatically.
Largest angle: the biggest angle is opposite the longest side — a quick way to pick which angle to find.

Worked examples

Example 1

A triangle has sides $b = 5\text{ cm}$ and $c = 7\text{ cm}$ , with included angle $A = 60^\circ$ . Find side $a$ .

Step 1: Substitute into the formula:

a^2 = 5^2 + 7^2 - 2(5)(7)\cos(60^\circ)

Step 2: Calculate:

a^2 = 25 + 49 - 70(0.5) = 74 - 35 = 39

Answer: $a = \sqrt{39} \approx 6.24\text{ cm}$ .

Example 2

A triangle has sides $a = 4\text{ cm}$ , $b = 5\text{ cm}$ , and $c = 6\text{ cm}$ . Calculate angle $A$ .

Step 1: Substitute into the rearranged formula:

\cos A = \frac{5^2 + 6^2 - 4^2}{2(5)(6)}

Step 2: Simplify:

\cos A = \frac{25 + 36 - 16}{60} = \frac{45}{60} = 0.75

Answer: $A = \cos^{-1}(0.75) \approx 41.4^\circ$ .

Example 3

A triangle has sides 8 m, 10 m, and 14 m. Find the size of the largest angle.

The largest angle is always opposite the largest side (14 m).

Step 1: Substitute into the formula for the angle opposite the 14 m side:

\cos \theta = \frac{8^2 + 10^2 - 14^2}{2(8)(10)}

Step 2: Simplify:

\cos \theta = \frac{64 + 100 - 196}{160} = \frac{-32}{160} = -0.2

Step 3: Solve for the angle:

Answer: $\theta = \cos^{-1}(-0.2) \approx 101.5^\circ$ .

Ready to practise?

11 Cosine Rule questions with answers and video solutions