Sine and Cosine Rules
The sine and cosine rules extend trigonometry to triangles that do not contain a right angle.
▶ Interact
Play with the controls to see the idea in action.
📘 Learn
The sine and cosine rules extend trigonometry to triangles that do not contain a right angle.
- Sine rule links sides to opposite angles
- Cosine rule is a squared equals b squared plus c squared minus 2bc cos A
- Use the cosine rule when you know two sides and the included angle
🧠 Quick quiz
Three questions. Score 2 or more to complete the topic.
What is Sine and Cosine Rules?
Within Mathematics, which covers numbers, algebra, geometry, trigonometry and statistics from first principles to exam level, this topic explains Sine and Cosine Rules. Maths builds step by step, so each topic here gives you a rule, a picture and a quick check you can use on homework and in exams.
Check that you can explain Sine and Cosine Rules in your own words and give one example from everyday life. If you can do that, you have understood the core idea.
Understanding Sine and Cosine Rules gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
Sine and Cosine Rules in detail
For triangles that are not right-angled, with sides a, b, c opposite angles A, B, C, the sine rule states a/sin A = b/sin B = c/sin C. Use it when you know an angle and its opposite side plus one more piece of information.
The cosine rule, a² = b² + c² − 2bc cos A, is used when you know two sides and the included angle, or all three sides (rearranged to find an angle). The ambiguous case with the sine rule can give two possible triangles. These rules solve problems in surveying, navigation and engineering.
How the interactive helps you understand Sine and Cosine Rules
Watch the triangle in the interactive. Use any sliders or buttons it offers to change the values and see how the triangle responds.
Before you change anything, predict the result. Then change one value at a time and check whether your prediction was right. This is a quick way to build a correct mental model of Sine and Cosine Rules.
Real-life uses and examples
- Use Sine and Cosine Rules when comparing prices, discounts or recipe quantities in everyday life.
- Explain Sine and Cosine Rules to a friend with a simple picture, such as a row of tiles or a bag of sweets.
- Spot Sine and Cosine Rules in exam questions, where it often sits inside a short word problem.
Common mistakes to avoid
- Forgetting units, signs or brackets when you apply Sine and Cosine Rules.
- Memorising the steps for Sine and Cosine Rules without checking why they work.
- Learning the definition without testing it. Use the quiz and the interactive to check that you can apply the idea.
Frequently asked questions about Sine and Cosine Rules
What is Sine and Cosine Rules?
The sine and cosine rules extend trigonometry to triangles that do not contain a right angle.
What are the most important points about Sine and Cosine Rules?
The main points are: Sine rule links sides to opposite angles; Cosine rule is a squared equals b squared plus c squared minus 2bc cos A; Use the cosine rule when you know two sides and the included angle.
Why is Sine and Cosine Rules useful?
Understanding Sine and Cosine Rules gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
How can I practise Sine and Cosine Rules?
Open the interactive at the top of this page and change the controls one at a time. Predict the result before each change, then take the three-question quiz. A score of two or more marks the topic as complete on this device.
Which topics are related to Sine and Cosine Rules?
Related topics in Mathematics include Bearings, Surface Area, Transformations: Translation and Reflection.