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24 Jan 2026
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Understanding Cosine and Sine Rules with Proofs - GCSE/IGCSE Guide
Neil Trivedi - MyEdSpace
@neildoesmaths
1 / 6
GCSE Maths: Cosine and Sine Rule Study Notes
You're about to master two essential formulas that'll make triangle problems much easier to solve. The cosine rule and sine rule are fundamental tools in GCSE maths that work with any triangle - not just right-angled ones.
These rules are particularly useful when you need to find missing measurements in triangles where basic SOH CAH TOA won't help. You'll use them in exams, coursework, and real-world applications like navigation and engineering.
Quick Tip: Think of these rules as upgraded versions of Pythagoras' theorem that work for all triangles!
The Cosine Rule - Finding Missing Sides and Angles
The cosine rule is your go-to method when you know two sides and the angle between them, or when you know all three sides but need an angle. It's like Pythagoras' theorem with an extra twist to handle non-right triangles.
For finding a missing length, use: a² = b² + c² - 2bc cos A. Simply plug in your known values and solve - it's that straightforward! In the example, with sides of 8cm and 14cm and a 60° angle between them, the missing side works out to 2√37 cm.
When you need a missing angle, rearrange the formula to: cos A = ÷ 2bc. Then use the inverse cosine function on your calculator to find the angle.
Exam Tip: Always double-check which sides and angles you're given before choosing your formula version!
Cosine Rule Example - Finding Missing Angles
Finding missing angles with the cosine rule follows a clear pattern that you can master with practice. When you know all three side lengths, you can calculate any angle using the rearranged cosine formula.
In this example with sides 5cm, 6cm, and 7cm, we substitute into cos A = (6² + 7² - 5²) ÷ (2 × 6 × 7). This gives us cos A = 5/7, so A = cos⁻¹(5/7) = 44.4°.
Remember to give your answer to the required number of significant figures - examiners always specify this! Your calculator's inverse cosine function (cos⁻¹) is essential here.
Memory Trick: The longest side is always opposite the largest angle, so use this to check if your answer makes sense!
The Sine Rule - When You Know Angles and Opposite Sides
The sine rule comes to the rescue when you know two sides and an angle that's not between them, or when you have two angles and one side. It creates a beautiful relationship: a/sin A = b/sin B = c/sin C.
For missing lengths, rearrange to solve for your unknown side. In the example, with a 45° angle, a 60° angle, and a 9cm side, you set up the equation and cross-multiply to get x = 3√6 cm.
The key insight is that you only need two parts of this three-way relationship to solve your problem. Pick the two fractions that contain your known and unknown values, then ignore the third part completely.
Study Smart: You can flip the sine rule to sin A/a = sin B/b = sin C/c when finding angles - whatever feels easier to work with!
Sine Rule for Angles - The Two-Step Method
Finding missing angles with the sine rule often requires a two-step approach that builds on the triangle angle sum. You'll first find one unknown angle, then use the fact that all triangle angles add up to 180°.
Step one involves using sin a = (sin 118° ÷ 21) × 13 to find angle a = 33.1°. This uses the rearranged sine rule where you multiply both sides to isolate the sine of your unknown angle.
Step two is simpler: since angles in a triangle sum to 180°, you calculate x = 180° - (118° + 33.1°) = 28.9°. This systematic approach prevents errors and ensures you don't miss any steps.
Confidence Booster: Break complex problems into smaller steps like this - you've got the skills to handle even tricky multi-step questions!
Proving the Sine Rule - Understanding Why It Works
Understanding the sine rule proof helps cement why this formula works and shows the elegant mathematics behind it. The proof uses the fact that triangle area can be calculated in multiple ways depending on which base and height you choose.
Starting with Area = ½bc sin A = ½ac sin B = ½ab sin C, you multiply everything by 2 to get bc sin A = ac sin B = ab sin C. This removes those pesky fractions and makes the next step clearer.
The magic happens when you divide everything by abc. This gives you (bc sin A) ÷ abc = (ac sin B) ÷ abc = (ab sin C) ÷ abc, which simplifies beautifully to sin A/a = sin B/b = sin C/c.
Deep Understanding: Proofs like this show that maths formulas aren't random - they're logical consequences of simpler ideas you already know!
We thought you’d never ask...
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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.
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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
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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 Knowunity AI. 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
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 Knowunity AI. 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
Understanding Cosine and Sine Rules with Proofs - GCSE/IGCSE Guide
Neil Trivedi - MyEdSpace
@neildoesmaths
Ever wondered how to solve triangles when you don't have a right angle to work with? The cosine rule and sine rule are your mathematical superpowers for tackling any triangle problem. These formulas help you find missing sides and angles... Show more
GCSE Maths: Cosine and Sine Rule Study Notes
You're about to master two essential formulas that'll make triangle problems much easier to solve. The cosine rule and sine rule are fundamental tools in GCSE maths that work with any triangle - not just right-angled ones.
These rules are particularly useful when you need to find missing measurements in triangles where basic SOH CAH TOA won't help. You'll use them in exams, coursework, and real-world applications like navigation and engineering.
Quick Tip: Think of these rules as upgraded versions of Pythagoras' theorem that work for all triangles!
Sign up to see the contentIt's free!
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
The Cosine Rule - Finding Missing Sides and Angles
The cosine rule is your go-to method when you know two sides and the angle between them, or when you know all three sides but need an angle. It's like Pythagoras' theorem with an extra twist to handle non-right triangles.
For finding a missing length, use: a² = b² + c² - 2bc cos A. Simply plug in your known values and solve - it's that straightforward! In the example, with sides of 8cm and 14cm and a 60° angle between them, the missing side works out to 2√37 cm.
When you need a missing angle, rearrange the formula to: cos A = ÷ 2bc. Then use the inverse cosine function on your calculator to find the angle.
Exam Tip: Always double-check which sides and angles you're given before choosing your formula version!
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
Cosine Rule Example - Finding Missing Angles
Finding missing angles with the cosine rule follows a clear pattern that you can master with practice. When you know all three side lengths, you can calculate any angle using the rearranged cosine formula.
In this example with sides 5cm, 6cm, and 7cm, we substitute into cos A = (6² + 7² - 5²) ÷ (2 × 6 × 7). This gives us cos A = 5/7, so A = cos⁻¹(5/7) = 44.4°.
Remember to give your answer to the required number of significant figures - examiners always specify this! Your calculator's inverse cosine function (cos⁻¹) is essential here.
Memory Trick: The longest side is always opposite the largest angle, so use this to check if your answer makes sense!
Sign up to see the contentIt's free!
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
The Sine Rule - When You Know Angles and Opposite Sides
The sine rule comes to the rescue when you know two sides and an angle that's not between them, or when you have two angles and one side. It creates a beautiful relationship: a/sin A = b/sin B = c/sin C.
For missing lengths, rearrange to solve for your unknown side. In the example, with a 45° angle, a 60° angle, and a 9cm side, you set up the equation and cross-multiply to get x = 3√6 cm.
The key insight is that you only need two parts of this three-way relationship to solve your problem. Pick the two fractions that contain your known and unknown values, then ignore the third part completely.
Study Smart: You can flip the sine rule to sin A/a = sin B/b = sin C/c when finding angles - whatever feels easier to work with!
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
Sine Rule for Angles - The Two-Step Method
Finding missing angles with the sine rule often requires a two-step approach that builds on the triangle angle sum. You'll first find one unknown angle, then use the fact that all triangle angles add up to 180°.
Step one involves using sin a = (sin 118° ÷ 21) × 13 to find angle a = 33.1°. This uses the rearranged sine rule where you multiply both sides to isolate the sine of your unknown angle.
Step two is simpler: since angles in a triangle sum to 180°, you calculate x = 180° - (118° + 33.1°) = 28.9°. This systematic approach prevents errors and ensures you don't miss any steps.
Confidence Booster: Break complex problems into smaller steps like this - you've got the skills to handle even tricky multi-step questions!
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
Proving the Sine Rule - Understanding Why It Works
Understanding the sine rule proof helps cement why this formula works and shows the elegant mathematics behind it. The proof uses the fact that triangle area can be calculated in multiple ways depending on which base and height you choose.
Starting with Area = ½bc sin A = ½ac sin B = ½ab sin C, you multiply everything by 2 to get bc sin A = ac sin B = ab sin C. This removes those pesky fractions and makes the next step clearer.
The magic happens when you divide everything by abc. This gives you (bc sin A) ÷ abc = (ac sin B) ÷ abc = (ab sin C) ÷ abc, which simplifies beautifully to sin A/a = sin B/b = sin C/c.
Deep Understanding: Proofs like this show that maths formulas aren't random - they're logical consequences of simpler ideas you already know!
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.
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4.9/5
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4.8/5
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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 Knowunity AI. 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
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 Knowunity AI. 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