•

31 Dec 2025

•

4 pages

Trigonometry Sine and Cosine Rules for Non-Calculator Problems

Emilia Wuss

@xed7even

The sine and cosine rules are powerful tools for solving... Show more

1 / 4

$The sime rude A A b CA Sides $ \frac{a}{Sin A} = \frac{b}{Sin B} = \frac{c}{Sin C}$ angle $\Rightarrow \frac{Sin A}{a} = \frac{Sin B}{b$

The Sine Rule

Ever wondered how to solve triangles that aren't right-angled? The sine rule is your go-to formula when dealing with any triangle where you know some sides and angles.

The sine rule states: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ . This can also be flipped to: $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$ .

Looking at the example: to find side 'a' when you know angle A = 37°, side b = 7.9, and angle B = 83°, you substitute into the formula. You get $a = \frac{7.9 \times \sin(37°)}{\sin(83°)}$ , which gives you the missing side length.

Remember: Use the sine rule when you have two angles and one side, or two sides and a non-included angle.

$The sime rude A A b CA Sides $ \frac{a}{Sin A} = \frac{b}{Sin B} = \frac{c}{Sin C}$ angle $\Rightarrow \frac{Sin A}{a} = \frac{Sin B}{b$

When to Use the Sine Rule

The sine rule works brilliantly in specific situations, and knowing when to use it will save you loads of time in exams.

Use the sine rule when you have two angles and one side (AAS or ASA triangles), or when you have two sides and an angle opposite one of them (SSA). The example shows angle A = 29°, side a = 19.2cm, and side c = 17.1cm.

Set up the equation: $\frac{\sin(29°)}{19.2} = \frac{\sin(C)}{17.1}$ . Cross-multiply to find the unknown angle C. This method is straightforward once you've identified which parts of the triangle you know.

Top tip: Always label your triangle clearly before starting - it prevents mix-ups with sides and angles!

$The sime rude A A b CA Sides $ \frac{a}{Sin A} = \frac{b}{Sin B} = \frac{c}{Sin C}$ angle $\Rightarrow \frac{Sin A}{a} = \frac{Sin B}{b$

The Cosine Rule - When to Use It

Sometimes the sine rule just won't cut it, and that's where the cosine rule becomes your best mate in triangle problems.

Use the cosine rule when you have a non-right-angled triangle and need to find a length with two known lengths plus the angle between them. You'll also use it when you know all three sides and need to find an angle.

The cosine rule is perfect for those tricky triangles where the sine rule leaves you stuck. If you've got two sides and the included angle, or three sides with no angles, the cosine rule is your answer.

Quick check: If you can't use the sine rule, try the cosine rule - one of them will always work!

$The sime rude A A b CA Sides $ \frac{a}{Sin A} = \frac{b}{Sin B} = \frac{c}{Sin C}$ angle $\Rightarrow \frac{Sin A}{a} = \frac{Sin B}{b$

The Cosine Rule Formula

The cosine rule formula is: $a^2 = b^2 + c^2 - 2bc \cos A$ . Notice how it's similar to Pythagoras' theorem but with an extra bit tacked on.

This formula helps you find the length of side 'a' when you know the other two sides (b and c) and the angle opposite to side 'a' (angle A). You can rearrange this formula for different sides depending on what you're looking for.

The cosine rule is incredibly reliable and works for any triangle - acute, obtuse, or right-angled. When the angle is 90°, cos(90°) = 0, so the formula actually becomes Pythagoras' theorem!

Pro tip: The cosine rule is your safety net - it works when other methods fail, making it perfect for exam confidence.

We thought you’d never ask...

What is the Knowunity AI companion?

Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.

Where can I download the Knowunity app?

You can download the app from Google Play Store and Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Most popular content in Maths

Comprehensive Maths Concepts

Explore essential mathematical concepts including powers, geometry, statistics, and probability. This resource features 65 pages of detailed explanations, diagrams, and examples to enhance your understanding of topics such as right triangles, volume calculations, and data representation. Ideal for students seeking to strengthen their numeracy skills and grasp complex mathematical principles.

Maths

GCSE maths notes

Pearson Edexcel

Maths

GCSE Higher Maths Essentials

Master key concepts in GCSE Higher Maths with this comprehensive summary covering geometry, algebra, probability, and circle theorems. Perfect for exam preparation, this resource includes essential formulas, angle facts, area calculations, and statistical measures to boost your understanding and performance. Ideal for students aiming for top grades.

Maths

Pythagoras theorem - foundation maths

Maths

Year 8 Math Concepts

Explore essential Year 8 math concepts including sequences, indices, area calculations, probability, and data representation. This knowledge organizer covers key topics such as central tendency, geometric properties, and algebraic manipulation, providing a comprehensive overview for effective study and understanding. Ideal for students preparing for assessments and seeking to strengthen their math skills.

Maths

Comprehensive Maths Concepts

Explore essential mathematical concepts including polynomial theorems, logarithmic properties, trigonometric functions, and integration techniques. This resource covers everything from solving inequalities to understanding exponential functions, providing a solid foundation for A-level mathematics. Ideal for students aiming for top grades.

Maths

Comprehensive Pure Mathematics

Explore detailed notes on Pure Mathematics Year 1, covering essential topics such as simultaneous equations, differentiation, integration, trigonometric functions, and more. Perfect for A Level Edexcel students seeking a thorough understanding of mathematical concepts and techniques.

Maths

Comprehensive Maths Concepts

Explore essential higher mathematics concepts including calculus, trigonometry, polynomials, and vector analysis. This summary covers key topics such as differentiation, integration, quadratic equations, and the properties of circles, providing a solid foundation for exam preparation. Ideal for students seeking a concise yet thorough review of advanced mathematical principles.

Maths

GCSE Maths (Higher) // Revision Guide

The only GCSE maths (higher) revision guide you need to get a grade 9! Contains every topic, each with all potential question types and their solutions.

Maths

Most popular content

An Inspector Calls: Character Insights

Explore in-depth analysis and key quotes for characters in J.B. Priestley's 'An Inspector Calls'. This resource covers Gerald Croft, Inspector Goole, Sheila Birling, Mrs. Birling, Eric Birling, and Eva Smith, focusing on themes of class, gender roles, and social responsibility. Ideal for students aiming for Grade 8 and above.

English Literature

aqa higher tier combined science - biology b1 : cell biology, paper 1

b1: cell biology - paper 1

Biology

Sociology of Families: Comprehensive Revision

Dive into an extensive overview of family dynamics, perspectives, and patterns in sociology. This resource covers key concepts such as family diversity, gender roles, marriage, and the impact of social policies on family structures. Perfect for A-Level Sociology students preparing for Paper 2.

Sociology

Comprehensive Crime & Deviance Overview

Explore an extensive revision of crime and deviance topics, including theories, types of crime, and the impact of media. This resource covers key concepts such as Marxism, functionalism, gender and crime, and the influence of globalization on criminal behavior. Ideal for students seeking a thorough understanding of criminology and its various theories. Type: Full Topic Revision.

Sociology

Sociology of Education Overview

Explore comprehensive A-Level Sociology notes on the education system, covering key theories, policies, and sociological perspectives. This resource includes insights on marketisation, gender roles, cultural deprivation, and educational inequalities, providing a thorough understanding of how education shapes social stratification and individual achievement. Ideal for exam preparation and in-depth study.

Sociology

Criminal Justice Evidence Rules

Explore the essential rules governing the use of evidence in criminal cases, including reliability, admissibility, and relevance. This summary covers key concepts such as the roles of personnel in investigations, the impact of witness testimonies, and the implications of plea bargaining. Ideal for Year 13 criminology students preparing for assessments.

Criminology

Biology Paper 1 Overview

Comprehensive study notes covering key concepts in cellular biology, human digestion, respiration, photosynthesis, and the circulatory system. This resource includes detailed explanations of cell structures, enzyme functions, nutrient absorption, and the impact of environmental factors on biological processes. Ideal for students preparing for Biology Paper 1 exams.

Biology

Macbeth: Guilt and Ambition

Explore the complex themes of guilt and ambition in Shakespeare's 'Macbeth'. This analysis covers key characters, including Macbeth and Lady Macbeth, their moral dilemmas, and the tragic consequences of their ambition. Ideal for students studying character motivations, thematic elements, and the psychological impact of power. Includes insights on the natural order, manipulation, and the descent into madness.

English Literature

Language Paper 1 Strategies

Master the AQA English Language Paper 1 with this comprehensive guide. Explore key strategies for language and structural analysis, critical evaluation, and creative writing. Learn how to effectively analyze texts, utilize literary techniques, and enhance your writing skills to excel in your exams.

English Language

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

Best app on earth! no words because it’s too good

Thomas R

iOS user

Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.

Basil

Android user

This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.

Rohan U

Android user

I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.

Xander S

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now

Paul T

iOS user

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Best app on earth! no words because it’s too good

Thomas R

iOS user

Basil

Android user

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Rohan U

Android user

Xander S

iOS user

Elisha

iOS user

Paul T

iOS user

Maths

•

326

•

31 Dec 2025

•

4 pages

Trigonometry Sine and Cosine Rules for Non-Calculator Problems

Emilia Wuss

@xed7even

The sine and cosine rules are powerful tools for solving non-right-angled triangles when you don't have a 90° angle to work with. These formulas help you find missing sides and angles in any triangle - skills you'll definitely need for... Show more

$The sime rude A A b CA Sides $ \frac{a}{Sin A} = \frac{b}{Sin B} = \frac{c}{Sin C}$ angle $\Rightarrow \frac{Sin A}{a} = \frac{Sin B}{b$

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

The Sine Rule

Ever wondered how to solve triangles that aren't right-angled? The sine rule is your go-to formula when dealing with any triangle where you know some sides and angles.

The sine rule states: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ . This can also be flipped to: $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$ .

Remember: Use the sine rule when you have two angles and one side, or two sides and a non-included angle.

$The sime rude A A b CA Sides $ \frac{a}{Sin A} = \frac{b}{Sin B} = \frac{c}{Sin C}$ angle $\Rightarrow \frac{Sin A}{a} = \frac{Sin B}{b$

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

When to Use the Sine Rule

The sine rule works brilliantly in specific situations, and knowing when to use it will save you loads of time in exams.

Top tip: Always label your triangle clearly before starting - it prevents mix-ups with sides and angles!

$The sime rude A A b CA Sides $ \frac{a}{Sin A} = \frac{b}{Sin B} = \frac{c}{Sin C}$ angle $\Rightarrow \frac{Sin A}{a} = \frac{Sin B}{b$

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

The Cosine Rule - When to Use It

Sometimes the sine rule just won't cut it, and that's where the cosine rule becomes your best mate in triangle problems.

Quick check: If you can't use the sine rule, try the cosine rule - one of them will always work!

$The sime rude A A b CA Sides $ \frac{a}{Sin A} = \frac{b}{Sin B} = \frac{c}{Sin C}$ angle $\Rightarrow \frac{Sin A}{a} = \frac{Sin B}{b$

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

The Cosine Rule Formula

The cosine rule formula is: $a^2 = b^2 + c^2 - 2bc \cos A$ . Notice how it's similar to Pythagoras' theorem but with an extra bit tacked on.

The cosine rule is incredibly reliable and works for any triangle - acute, obtuse, or right-angled. When the angle is 90°, cos(90°) = 0, so the formula actually becomes Pythagoras' theorem!

Pro tip: The cosine rule is your safety net - it works when other methods fail, making it perfect for exam confidence.