Subjects
Biology
Chemistry
AP Chemistry
English
AP English Literature
Geometry
Dec 4, 2025
54
6 pages
Neil Trivedi - MyEdSpace @neildoesmaths
Ever wondered how to solve triangles when you don't have a right angle to work with? The cosine... Show more
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 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!
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 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!
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!
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!
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
2
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
App Store
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
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
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
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!
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 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!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
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!
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 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!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
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!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
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!
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
2
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
App Store
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
