The Sine and Cosine Rules Revision Notes for AQA GCSE Maths

The Sine and Cosine Rules

Introduction to sine and cosine rules

The basic trigonometric ratios (sine, cosine, tangent) that you learned with SOH CAH TOA only work for right-angled triangles. But what happens when you need to work with triangles that don't have a right angle? This is where the sine and cosine rules come to the rescue, allowing you to solve problems involving any triangle.

These rules are essential tools that extend your trigonometric toolkit beyond the limitations of right-angled triangles, opening up a whole new world of problem-solving possibilities.

Labelling triangles correctly

Before you can use these rules effectively, you must label your triangles properly. This step cannot be overlooked as it forms the foundation for all subsequent calculations.

chatImportant

Triangle Labelling is Absolutely Crucial

Here's how to do it correctly:

Use lowercase letters (a, b, c) for the sides of the triangle
Use uppercase letters (A, B, C) for the angles
Make sure that each side and its opposite angle correspond to each other

The key principle: side 'a' is opposite angle A, side 'b' is opposite angle B, and side 'c' is opposite angle C. It doesn't matter which sides you decide to call a, b, and c, as long as you label the angles correctly to match.

The three essential formulas

There are three main formulas you need to master for working with any triangle. Each formula serves a specific purpose and knowing when to use each one is crucial for success.

The sine rule

The sine rule states that the ratio of any side to the sine of its opposite angle is constant for all sides of a triangle:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

You don't need to use all three parts of this equation at once. Instead, you choose the two parts that contain the information you have and the value you want to find. For example, you might use:

$\frac{b}{\sin B} = \frac{c}{\sin C}$ , or
$\frac{a}{\sin A} = \frac{b}{\sin B}$

The cosine rule

The cosine rule comes in two forms, depending on what you're trying to find:

For finding a side: $a^2 = b^2 + c^2 - 2bc \cos A$

For finding an angle: $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$

The second form is simply a rearrangement of the first formula, making it easier to find angles when you know all three sides.

Area of a triangle

When you know two sides and the angle between them, you can find the area using:

$\text{Area} = \frac{1}{2}ab \sin C$

This formula is particularly useful because it works for any triangle, not just right-angled ones.

The four key scenarios

Understanding when to use each rule is crucial. There are four main types of problems you'll encounter, and recognising these patterns will help you choose the correct approach every time.

infoNote

The Four Key Scenarios for Applying Sine and Cosine Rules:

Scenario 1: Two angles and any side → Use sine rule

When you're given two angles and any side of a triangle, you can use the sine rule to find the remaining sides
First calculate the third angle (since angles in a triangle sum to 180°), then apply the sine rule

Scenario 2: Two sides and an angle not enclosed by them → Use sine rule

If you have two sides and an angle that is not between those two sides, the sine rule is your tool of choice
You can use this information to find the remaining angles and sides

Scenario 3: Two sides and the enclosed angle → Use cosine rule

When you know two sides and the angle between them, you need the cosine rule to find the third side
This is one of the most common applications of the cosine rule

Scenario 4: All three sides, no angles → Use cosine rule

If you know all three sides of a triangle but no angles, you can use the rearranged cosine rule to find any of the angles
This is the only way to find angles when you only have side lengths

Working through problems systematically

When tackling trigonometry problems, a structured approach will help you avoid errors and work more efficiently.

infoNote

Step-by-Step Problem-Solving Approach:

Label your triangle properly - This prevents confusion and errors
Identify what you know and what you need to find - This helps you choose the right rule
Select the appropriate formula - Use the four scenarios above as your guide
Substitute your values carefully - Double-check that you're using the right numbers
Calculate step by step - Show your working clearly
Check your answer makes sense - Does the result seem reasonable?

Common mistakes to avoid

Even with a good understanding of the rules, there are several pitfalls that can trip you up. Being aware of these common errors will help you avoid them.

chatImportant

Critical Mistakes to Avoid:

Forgetting to label triangles correctly before starting calculations
Using the sine rule when you should use the cosine rule (and vice versa)
Mixing up which sides and angles correspond to each other
Not checking whether your calculator is in degree or radian mode
Rounding too early in multi-step calculations

These mistakes can completely derail your solution, so always double-check these fundamental aspects before proceeding with calculations.

bookmarkSummary

Key Points to Remember:

The sine and cosine rules extend trigonometry beyond right-angled triangles to work with any triangle
Proper triangle labelling is essential - side 'a' must be opposite angle A, and so on
Use the sine rule when you have two angles and any side, or two sides and an angle not between them
Use the cosine rule when you have two sides and the enclosed angle, or when you have all three sides
The area formula $\left(\frac{1}{2}ab \sin C\right)$ is useful when you know two sides and the included angle
Always identify which scenario you're dealing with before selecting your formula

The Sine and Cosine Rules (AQA GCSE Maths): Revision Notes