The cosine rule and sine rule are your go-to formulas... Show more
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rue ๐
11/12/2025
Maths
maths sine and cosine rule
379
โข
11 Dec 2025
โข
Understanding the Sine and Cosine Rules in Maths
rue ๐
@lanastudygrlxo
1 / 3
Cosine Rule
When you're dealing with a triangle that's not right-angled, the cosine rule becomes your best mate. You'll use it when you know two sides and the angle between them, or when you know all three sides.
The formula is aยฒ = bยฒ + cยฒ - 2bc cos A. If you need to find an angle instead, rearrange it to cos A = / 2bc. Notice how it's like Pythagoras' theorem but with that extra bit tagged on the end.
Let's say you've got a triangle with sides 5cm and 10cm, and the angle between them is what you're after. If the opposite side is 9cm, you'd substitute into the rearranged formula. Work through: cos A = (5ยฒ + 10ยฒ - 9ยฒ) / (2ร5ร10) = 44/100 = 0.44, then use cosโปยน(0.44) to get 63.9ยฐ.
Quick tip: Remember to use the cosโปยน function on your calculator to convert back from the cosine value to the actual angle!
Sine Rule
The sine rule is brilliant for finding unknown angles or side lengths in non-right-angled triangles. It's especially handy when you know two angles and one side, or two sides and an angle opposite one of them.
The formula is a/sin A = b/sin B = c/sin C. Each side divided by the sine of its opposite angle gives the same value - pretty neat! You can rearrange this to find whatever you're looking for.
Here's how it works: if you've got a 6cm side opposite a 40ยฐ angle, and a 7cm side opposite angle C, set up 6/sin 40ยฐ = 7/sin C. Cross multiply to get sin C = (7 ร sin 40ยฐ)/6 = 0.750. Then use sinโปยน to find C = 48.6ยฐ.
Watch out: Always remember to use the inverse sine function (sinโปยน) to convert your decimal answer back to an angle - don't forget this crucial final step!
Practice Problem
Time to put your skills to the test! In triangle ABC, you've got angle A = 44ยฐ, angle C = 64ยฐ, and side BC = 4cm. Your mission is to find the length of AB and calculate the triangle's area.
Start by working out the missing angle B - remember all angles in a triangle add up to 180ยฐ. So angle B = 180ยฐ - 44ยฐ - 64ยฐ = 72ยฐ.
Now you can use the sine rule to find AB. Set up your equation: AB/sin C = BC/sin A, which gives you AB/sin 64ยฐ = 4/sin 44ยฐ. Cross multiply and solve to get AB to 3 decimal places.
For the area, use the formula Area = ยฝab sin C, where a and b are two sides and C is the angle between them. You'll have all the values you need once you've found AB!
Pro tip: Always start by finding any missing angles first - it often makes the rest of the problem much easier to tackle!
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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 itโs 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 didnโt even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as Iโm 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 ๐๐๐ฒ๐ค๐โจ๐๐ฎ
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 itโs 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 didnโt even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as Iโm 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 ๐๐๐ฒ๐ค๐โจ๐๐ฎ
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 the Sine and Cosine Rules in Maths
rue ๐
@lanastudygrlxo
The cosine rule and sine rule are your go-to formulas for solving any triangle - even when there's no right angle in sight! These powerful tools help you find missing sides and angles when you've got enough information to work... Show more
Cosine Rule
When you're dealing with a triangle that's not right-angled, the cosine rule becomes your best mate. You'll use it when you know two sides and the angle between them, or when you know all three sides.
The formula is aยฒ = bยฒ + cยฒ - 2bc cos A. If you need to find an angle instead, rearrange it to cos A = / 2bc. Notice how it's like Pythagoras' theorem but with that extra bit tagged on the end.
Let's say you've got a triangle with sides 5cm and 10cm, and the angle between them is what you're after. If the opposite side is 9cm, you'd substitute into the rearranged formula. Work through: cos A = (5ยฒ + 10ยฒ - 9ยฒ) / (2ร5ร10) = 44/100 = 0.44, then use cosโปยน(0.44) to get 63.9ยฐ.
Quick tip: Remember to use the cosโปยน function on your calculator to convert back from the cosine value to the actual angle!
Sign up to see the contentIt's free!
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Improve your grades
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Sine Rule
The sine rule is brilliant for finding unknown angles or side lengths in non-right-angled triangles. It's especially handy when you know two angles and one side, or two sides and an angle opposite one of them.
The formula is a/sin A = b/sin B = c/sin C. Each side divided by the sine of its opposite angle gives the same value - pretty neat! You can rearrange this to find whatever you're looking for.
Here's how it works: if you've got a 6cm side opposite a 40ยฐ angle, and a 7cm side opposite angle C, set up 6/sin 40ยฐ = 7/sin C. Cross multiply to get sin C = (7 ร sin 40ยฐ)/6 = 0.750. Then use sinโปยน to find C = 48.6ยฐ.
Watch out: Always remember to use the inverse sine function (sinโปยน) to convert your decimal answer back to an angle - don't forget this crucial final step!
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
Practice Problem
Time to put your skills to the test! In triangle ABC, you've got angle A = 44ยฐ, angle C = 64ยฐ, and side BC = 4cm. Your mission is to find the length of AB and calculate the triangle's area.
Start by working out the missing angle B - remember all angles in a triangle add up to 180ยฐ. So angle B = 180ยฐ - 44ยฐ - 64ยฐ = 72ยฐ.
Now you can use the sine rule to find AB. Set up your equation: AB/sin C = BC/sin A, which gives you AB/sin 64ยฐ = 4/sin 44ยฐ. Cross multiply and solve to get AB to 3 decimal places.
For the area, use the formula Area = ยฝab sin C, where a and b are two sides and C is the angle between them. You'll have all the values you need once you've found AB!
Pro tip: Always start by finding any missing angles first - it often makes the rest of the problem much easier to tackle!
We thought youโd 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 โ all at your fingertips.
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Most popular content in Maths
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Can't find what you're looking for? Explore other subjects.
Students love us โ 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 itโs 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 didnโt even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as Iโm 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 ๐๐๐ฒ๐ค๐โจ๐๐ฎ
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 itโs 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 didnโt even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as Iโm 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 ๐๐๐ฒ๐ค๐โจ๐๐ฎ
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