Use the Sine and cosine rules - GCSE Maths, Higher

The sine and cosine rules are used to find the missing sides or angles in triangles that are not right-angled. They are used instead of Pythagoras and basic trigonometry, which don't apply to these types of triangles.

The shorthand for these rules is cos and sin. They are a feature of the higher-tier of GCSE Maths and should be expected in exams.

This article will explain the cosine and sine rules. It will explain the formula and how it works, including providing examples. Quiz questions are included to test your understanding. It is suitable for GCSE Maths revision across all major exam boards.

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Sine rule - Formula

The sine rule is a formula for finding missing angles or sides in a triangle that isn't right-angled. It links each side to the sine, which is a trigonometric value based on the size of the angle opposite it.

This is the formula:

a / sin A = b / sin B = c / sin C

Each letter represents a matching pair. The pairing itself is important, as every side must match the angle that is directly opposite:

Exam questions will typically expect you to compare two fractions at a time, using the sine rule. For instance, you may be asked to compare side "a" with side "b".

When using the sine rule, you normally need to substitute known values into the formula. You will also need to rearrange the equation. It's advisable to use a calculator in degree mode for accuracy.

When to use the Sine rule

You should use the sine rule when the triangle is not right-angled, and you know one of the angles and the side opposite it. The sine rule is built around matching sides to their opposite angles, so you must have an opposite pairing.

In exams, you will apply the sine rule when there are two angles and one side known. You need to find the other side. Alternatively, you may be given two known sides and an angle, provided that the known angle is opposite one of the known sides.

If you don't have an angle-opposite pairing, the sine rule is not appropriate for calculations. You will need to use another method, such as the cosine rule.

The sine rule isn't limited to a specific triangle shape. As long as the triangle isn't right-angled and you have the correct information, the rule can be applied.

Cosine Rule - Formula

The cosine rule is used to find a missing side or angle when the triangle doesn't have a right angle. Because of this, it's considered an extension of the Pythagoras theorem.

This is the formula:

a² = b² + c² - 2bc cos A

This is the breakdown of the formula. The letters can be changed depending on the side or angle you are trying to find:

To use the cosine rule, the triangle must have two known sides and a known angle between them. Alternatively, you can find an angle with the cos rule when all three sides of the triangle are given.

When to use the Cosine rule

The triangle needs to have two known sides, plus the angle between them. You need to find the third side. Alternatively, all three sides must be known if you're asked to find an angle. You don't need to have an opposite pairing, unlike the sine rule.

This rule works with squares of length. If the triangle isn't right-angled and the question resembles Pythagoras’ theorem, it's a good idea to use the cos rule.

Remember that this formula applies to the side or angle opposite what you are trying to find. If the angle and side don't match, this calculation will fail.

Worked Examples - Find the angle and sides of a triangle

The questions we have given below are typical for GCSE Maths. Trigonometry calculations are typically visual questions, and you should expect to work with a diagram.

Example 1 - Finding a Side (Sine)

The triangle shows the following:

The sine rule can be used here because the triangle has given an angle and its opposite side.

Workings:

Example 2 - Finding a side (Cosine)

The triangle shows the following:

The cosine rule can be used because two sides and an angle are known.

Workings:

Example 3 - Finding an angle (Cosine)

The triangle shows the following:

The cosine rule can be applied because all the sides are known, and we need to find angle A.

Workings:

Maths quiz - Practice sine and cosine rules

3

What method can find the length of side b?

Pythagoras’ theorem

The sine rule

The cosine rule

Correct

There is a known angle and the opposite side. This means the sine rule can be used.

Submit

4

What method can find the missing side?

The cosine rule

The sine rule

Basic trigonometry (Sine, Cosine, Tangent)

Correct

There are two known sides and the angle between them. This fits the requirements for the cosine rule.

Submit

Conclusion

The sine and cosine rules can be used to find angles and sides for triangles that aren't right-angled. The formulas show a clear relationship between sides and angles. Apply them carefully, substituting the triangle values into the formula, rearranging your workings, and simplifying as a final step.

The first step in tackling a geometry problem is to identify the information you're given. Then you can decide what the correct rule is, whether it's the sine rule, cosine rule, basic trigonometry, or Pythagoras' theorem. Show your workings clearly in logical steps, to avoid mistakes and earn more marks in your exams.

If this topic is complex and there are gaps in your understanding, you may benefit from an overview of right-angled triangles by BBC Bitesize. Alternatively, you can test your understanding with exam questions on the cosine rule and sine rule by Corbett Maths.

If you need support, TeachTutti has experienced Maths GCSE tutors. Every tutor has an enhanced DBS check and will tailor lessons to your specific needs, such as understanding the theory behind each mathematical technique and learning how to use them efficiently.