Coterminal angles have the same terminal side but different measures.... Show more
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Finding Coterminal Angles
Ever noticed how a car's steering wheel can turn multiple times but still position the wheels the same way? That's similar to coterminal angles! Coterminal angles are angles in standard position that share the same terminal side, regardless of their measure.
When angles exceed 360° or are negative, they can be awkward to work with. Fortunately, every angle has a corresponding angle between 0° and 360° that's easier to handle. For example, both 140° and -220° have the same terminal side, making them coterminal angles.
Any angle actually has infinitely many coterminal angles. Just add or subtract 360° (a complete rotation) to find them! For instance, 100°, 460°, and -260° are all coterminal because they share the same terminal side.
Quick Tip: When an angle seems too large or negative, find its coterminal angle between 0° and 360° by adding or subtracting multiples of 360° until you get within range.
Reference Angles and Coterminal Angles
Reference angles help us understand angles in all quadrants. A reference angle is the smallest positive acute angle formed between the terminal side of an angle and the x-axis. Think of it as the "shortcut" angle that tells you how far the terminal side is from the horizontal axis.
Reference angles change depending on the quadrant:
- Quadrant I: reference angle = angle
- Quadrant II: reference angle = 180° - angle
- Quadrant III: reference angle = angle - 180°
- Quadrant IV: reference angle = 360° - angle
Finding coterminal angles with measures greater than 360° is straightforward. For example, to find the coterminal angle for 800°, subtract 360° until you get an angle between 0° and 360°: 800° - 360° = 440°, then 440° - 360° = 80°. So 80° is coterminal with 800°.
Visualize It: 800° equals 80° plus two complete rotations (720°) around the circle. Both angles end at exactly the same position!
Finding Coterminal Angles with Negative Measures
Negative angles aren't as scary as they seem! When you see a negative angle like -45°, just remember it means you're moving clockwise instead of counterclockwise around the circle.
To find a positive coterminal angle for -45°, add 360°: -45° + 360° = 315°
Both -45° and 315° end at the same position on the circle, but 315° is within our preferred range of 0° to 360°. When visualized, -45° is measured clockwise from the positive x-axis, while 315° is measured counterclockwise.
You can apply this same technique to any negative angle. Just keep adding 360° until you get a positive angle in the range you want.
Remember: Negative angles go clockwise, positive angles go counterclockwise, but coterminal angles always end at the same point regardless of how you get there!
Working with Radians
Coterminal angles work the same way in radians as they do in degrees. The only difference is that instead of adding or subtracting 360°, you add or subtract 2π radians (a full rotation).
To find a coterminal angle for 19π/4 radians that's between 0 and 2π:
First, subtract 2π: 19π/4 - 2π = 19π/4 - 8π/4 = 11π/4
Since 11π/4 is still greater than 2π , subtract 2π again: 11π/4 - 2π = 11π/4 - 8π/4 = 3π/4
The angle 3π/4 is coterminal with 19π/4 and falls within our desired range!
Pro Tip: Convert fractions to the same denominator when working with radian measures. This makes it easier to add or subtract and see if you're in the correct range.
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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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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❗️❗️⚠️
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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
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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
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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
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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
Understanding and Calculating Coterminal Angles Easily
Coterminal angles have the same terminal side but different measures. Think of them as different paths that end at the same point on the unit circle. Understanding how to find coterminal angles helps simplify trigonometric problems and makes working with... Show more
Finding Coterminal Angles
Ever noticed how a car's steering wheel can turn multiple times but still position the wheels the same way? That's similar to coterminal angles! Coterminal angles are angles in standard position that share the same terminal side, regardless of their measure.
When angles exceed 360° or are negative, they can be awkward to work with. Fortunately, every angle has a corresponding angle between 0° and 360° that's easier to handle. For example, both 140° and -220° have the same terminal side, making them coterminal angles.
Any angle actually has infinitely many coterminal angles. Just add or subtract 360° (a complete rotation) to find them! For instance, 100°, 460°, and -260° are all coterminal because they share the same terminal side.
Quick Tip: When an angle seems too large or negative, find its coterminal angle between 0° and 360° by adding or subtracting multiples of 360° until you get within range.
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Reference Angles and Coterminal Angles
Reference angles help us understand angles in all quadrants. A reference angle is the smallest positive acute angle formed between the terminal side of an angle and the x-axis. Think of it as the "shortcut" angle that tells you how far the terminal side is from the horizontal axis.
Reference angles change depending on the quadrant:
- Quadrant I: reference angle = angle
- Quadrant II: reference angle = 180° - angle
- Quadrant III: reference angle = angle - 180°
- Quadrant IV: reference angle = 360° - angle
Finding coterminal angles with measures greater than 360° is straightforward. For example, to find the coterminal angle for 800°, subtract 360° until you get an angle between 0° and 360°: 800° - 360° = 440°, then 440° - 360° = 80°. So 80° is coterminal with 800°.
Visualize It: 800° equals 80° plus two complete rotations (720°) around the circle. Both angles end at exactly the same position!
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Finding Coterminal Angles with Negative Measures
Negative angles aren't as scary as they seem! When you see a negative angle like -45°, just remember it means you're moving clockwise instead of counterclockwise around the circle.
To find a positive coterminal angle for -45°, add 360°: -45° + 360° = 315°
Both -45° and 315° end at the same position on the circle, but 315° is within our preferred range of 0° to 360°. When visualized, -45° is measured clockwise from the positive x-axis, while 315° is measured counterclockwise.
You can apply this same technique to any negative angle. Just keep adding 360° until you get a positive angle in the range you want.
Remember: Negative angles go clockwise, positive angles go counterclockwise, but coterminal angles always end at the same point regardless of how you get there!
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Working with Radians
Coterminal angles work the same way in radians as they do in degrees. The only difference is that instead of adding or subtracting 360°, you add or subtract 2π radians (a full rotation).
To find a coterminal angle for 19π/4 radians that's between 0 and 2π:
First, subtract 2π: 19π/4 - 2π = 19π/4 - 8π/4 = 11π/4
Since 11π/4 is still greater than 2π , subtract 2π again: 11π/4 - 2π = 11π/4 - 8π/4 = 3π/4
The angle 3π/4 is coterminal with 19π/4 and falls within our desired range!
Pro Tip: Convert fractions to the same denominator when working with radian measures. This makes it easier to add or subtract and see if you're in the correct range.
We thought you’d never ask...
What is the Knowunity AI companion?
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.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the 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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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