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Cosine Rule: Formula, Calculator, Worked Examples & Law of Cosines Guide

Learn the cosine rule (also called the law of cosines) with clear formulas, step-by-step methods, a free calculator, GCSE-style examples, and exam tips for finding sides and angles in any triangle.

Quick Answer: What is the Cosine Rule?

The cosine rule is a triangle formula used when you know two sides and the included angle, or all three sides. It helps you find a missing side or angle in any triangle, not just right-angled triangles.

Find a side:

\[a^2 = b^2 + c^2 - 2bc\cos(A)\]

Find an angle:

\[\cos(A)=\frac{b^2+c^2-a^2}{2bc}\]

Best for: SAS and SSS triangle questions.

📑 Quick Navigation

→ What is the Cosine Rule? → Cosine Rule Formula → When to Use It → Cosine Rule Calculator → Step-by-Step Method → Worked Examples → Common Mistakes → Proof / Derivation → Sine Rule vs Cosine Rule → GCSE / IGCSE / IB Exam Tips → Practice Questions → FAQs

What is the Cosine Rule?

The cosine rule is one of the most important formulas in trigonometry. It is also known as the law of cosines. Students use it to solve triangles when the information given does not fit the sine rule. In simple terms, the cosine rule links the lengths of sides in a triangle to the cosine of one of its angles.

The reason the cosine rule is so useful is that it works for any triangle. You can use it for acute triangles, obtuse triangles, and even right-angled triangles. In fact, if one angle is exactly 90°, the cosine rule becomes the Pythagorean theorem because \(\cos(90^\circ)=0\). This makes it a powerful general formula rather than a narrow special-case rule.

In school mathematics, the cosine rule usually appears in GCSE, IGCSE, A-Level, and IB Math. It is often tested in exam questions where you must choose the correct method yourself. That is why understanding when to use it matters just as much as memorizing the formula.

🎯 Key Facts You Should Know

Also called: Law of cosines
Used for: Finding a missing side or angle
Best cases: SAS and SSS triangles
Works for: Any triangle
Exam importance: Common in trigonometry and geometry questions
Common confusion: Students often mix it up with the sine rule, so learning the difference is essential

If you want to solve triangle problems confidently, the cosine rule is non-negotiable. Once you know the formula, how to label the triangle, and how to identify the right scenario, it becomes much easier to solve even complex-looking questions.

Cosine Rule Formula

The cosine rule has two main forms. One is used to find a missing side. The other is used to find a missing angle. Both are based on the same relationship, but the second is rearranged to isolate the cosine of an angle.

📏 Formula for Finding a Side

Use this when you know two sides and the included angle:

\[a^2 = b^2 + c^2 - 2bc \cos(A)\]

Here, \(a\) is the side opposite angle \(A\). The known sides are \(b\) and \(c\), and \(A\) is the angle between them.

📐 Formula for Finding an Angle

Use this when you know all three sides:

\[\cos(A)=\frac{b^2+c^2-a^2}{2bc}\]

After finding \(\cos(A)\), use inverse cosine:

\[A=\cos^{-1}\left(\frac{b^2+c^2-a^2}{2bc}\right)\]

📋 Full Set of Cosine Rule Forms

Depending on which side or angle you are working with, you may see these forms:

\[a^2=b^2+c^2-2bc\cos(A)\]

\[b^2=a^2+c^2-2ac\cos(B)\]

\[c^2=a^2+b^2-2ab\cos(C)\]

When to Use the Cosine Rule

One of the biggest reasons students lose marks is not because they cannot calculate, but because they choose the wrong rule. The cosine rule should be your first thought in two cases: SAS and SSS.

Scenario 1: SAS

You know two sides and the included angle. This is the classic side-finding cosine-rule problem. The angle must be between the two known sides.

Scenario 2: SSS

You know all three sides. This is the angle-finding version. Rearrange the formula and use inverse cosine.

Decision Shortcut

Use the cosine rule if the triangle information looks like this:

Two sides + included angle → find the third side
Three sides → find an angle
You cannot match a side with its opposite angle pair for sine rule

🧮 Cosine Rule Calculator

Use this free cosine rule calculator to find a missing side or a missing angle. This is especially useful when checking homework, verifying exam working, or learning how the formula behaves with different triangles.

Choose Calculation Type:

Find Side Length (SAS)

Enter two sides and the included angle

Side b:

Side c:

Angle A (degrees):

How to Use the Cosine Rule Step by Step

The cosine rule becomes much easier once you follow a reliable process. Most errors happen because students rush the labeling or plug the wrong side opposite the wrong angle. The safest method is to slow down, label correctly, and write the formula before substituting any numbers.

📝 Finding a Side with the Cosine Rule

Step 1: Label the unknown side as the side opposite the known angle.

Step 2: Write the formula \(a^2=b^2+c^2-2bc\cos(A)\).

Step 3: Substitute the two side lengths and the included angle.

Step 4: Work out \(a^2\), then take the square root.

Step 5: Give the answer with correct units and suitable accuracy.

📝 Finding an Angle with the Cosine Rule

Step 1: Identify the side opposite the angle you want.

Step 2: Use \(\cos(A)=\frac{b^2+c^2-a^2}{2bc}\).

Step 3: Substitute all three side lengths carefully.

Step 4: Simplify to get a decimal value for \(\cos(A)\).

Step 5: Use inverse cosine to find the angle in degrees.

Cosine Rule Worked Examples

Worked examples are the fastest way to understand how the formula behaves in real questions. Below are side-finding and angle-finding examples, followed by a more advanced obtuse-angle example that students often find tricky in exams.

Example 1: Find a Side

In triangle ABC, \(b=7\) cm, \(c=9\) cm, and \(A=60^\circ\). Find \(a\).

\[ a^2 = 7^2 + 9^2 - 2(7)(9)\cos(60^\circ) \] \[ a^2 = 49 + 81 - 126(0.5) \] \[ a^2 = 130 - 63 = 67 \] \[ a = \sqrt{67} \approx 8.19 \text{ cm} \]

Example 2: Find an Angle

A triangle has sides \(a=8\), \(b=7\), \(c=9\). Find angle \(A\).

\[ \cos(A)=\frac{7^2+9^2-8^2}{2(7)(9)} \] \[ \cos(A)=\frac{49+81-64}{126} \] \[ \cos(A)=\frac{66}{126}=0.5238 \] \[ A=\cos^{-1}(0.5238)\approx 58.41^\circ \]

Example 3: Obtuse Triangle

A triangle has sides \(a=12\), \(b=5\), \(c=10\). Find angle \(A\).

\[ \cos(A)=\frac{5^2+10^2-12^2}{2(5)(10)} \] \[ \cos(A)=\frac{25+100-144}{100} \] \[ \cos(A)=\frac{-19}{100}=-0.19 \] \[ A=\cos^{-1}(-0.19)\approx 100.95^\circ \]

Because cosine is negative, the angle is obtuse. This is one reason the cosine rule is especially helpful: it handles obtuse triangles cleanly.

Common Cosine Rule Mistakes

Using the wrong angle: the angle in the formula must be opposite the side you label as \(a\).
Mixing up sine rule and cosine rule: use cosine rule for SAS and SSS, not because the formula “looks familiar.”
Forgetting the negative sign: the formula has - 2bc cos(A), not +.
Not taking the square root: when finding a side, you usually get \(a^2\) first, not \(a\).
Calculator mode errors: make sure your calculator is in degree mode if the angle is in degrees.
Rounding too early: keep full calculator values until the final line.
Using invalid side lengths: if the sides do not satisfy triangle inequality, no real triangle exists.

Proof of the Cosine Rule

Understanding where the cosine rule comes from helps you remember it better. One common proof starts by dropping a perpendicular from one vertex of a triangle and splitting the triangle into two right-angled triangles. Then Pythagoras and basic trigonometry are used to connect the pieces.

Suppose triangle ABC has sides \(a\), \(b\), and \(c\), with angle \(A\) opposite side \(a\). Drop a perpendicular from the vertex at angle \(A\). If one part of the base is \(c\cos(A)\) and the height is \(c\sin(A)\), then by applying Pythagoras to the right triangle:

\[ a^2 = (b - c\cos(A))^2 + (c\sin(A))^2 \]

Expand:

\[ a^2 = b^2 - 2bc\cos(A) + c^2\cos^2(A) + c^2\sin^2(A) \]

Use the identity \(\sin^2(A)+\cos^2(A)=1\):

\[ a^2 = b^2 + c^2 - 2bc\cos(A) \]

That is the cosine rule.

Sine Rule vs Cosine Rule

Many students know both formulas but hesitate when deciding which one to use. The easiest way to remember the difference is this: the sine rule needs an opposite angle-side pair, while the cosine rule does not.

Aspect	Sine Rule	Cosine Rule
Best used when	You know an opposite side-angle pair	You have SAS or SSS
Main idea	Ratios of sides and opposite sines	Connects sides and cosine of an angle
Formula style	\(\frac{a}{\sin A}=\frac{b}{\sin B}\)	\(a^2=b^2+c^2-2bc\cos A\)
Ambiguous case?	Yes, possible in SSA	No

GCSE, IGCSE, A-Level and IB Exam Tips

Write the formula before substituting — examiners often reward method marks.
Always sketch or label the triangle if one is not already provided.
State whether you are using degrees, especially if your calculator is shared or reset.
Do not round early; keep full decimals until the final answer.
Check whether the question is side-finding or angle-finding before choosing the version of the formula.
For harder questions, use cosine rule first, then sine rule after one missing part is found.
In word problems, identify whether the angle given is included between the known sides.

Practice Questions with Answers

Practice

A triangle has sides 6 cm and 11 cm with included angle \(45^\circ\). Find the third side.
A triangle has sides 10 cm, 12 cm, and 15 cm. Find the angle opposite the 10 cm side.
A triangle has sides 9 cm and 13 cm with included angle \(120^\circ\). Find the third side.
A triangle has sides 7 cm, 8 cm, and 12 cm. Find the angle opposite the 12 cm side.

Answers

1. Third side \(\approx 7.78\) cm
2. Angle \(\approx 41.41^\circ\)
3. Third side \(\approx 19.72\) cm
4. Angle \(\approx 110.49^\circ\)

Frequently Asked Questions About the Cosine Rule

What is the cosine rule?

The cosine rule is a formula used to find a side or an angle in any triangle. It is especially useful for SAS and SSS triangle problems.

What is the law of cosines?

The law of cosines is just another name for the cosine rule. Both terms refer to the same triangle formula.

When do you use the cosine rule?

Use the cosine rule when you know two sides and the included angle, or when you know all three sides and need to find an angle.

How do you use the cosine rule to find a side?

Substitute the two known side lengths and the included angle into \(a^2=b^2+c^2-2bc\cos(A)\), then take the square root.

How do you use the cosine rule to find an angle?

Rearrange to \(\cos(A)=\frac{b^2+c^2-a^2}{2bc}\), calculate the value, then use inverse cosine.

Does the cosine rule work for right-angled triangles?

Yes. If the angle is 90°, the cosine rule becomes the Pythagorean theorem because \(\cos 90^\circ = 0\).

What is the difference between sine rule and cosine rule?

The sine rule uses opposite angle-side pairs. The cosine rule is used for SAS and SSS cases and is better when no opposite angle-side pair is known.

Why do students lose marks on cosine-rule questions?

Most lost marks come from wrong labeling, using the wrong angle, forgetting the minus sign, or failing to take the square root at the end.

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