PRO Courses Hot Guides Tech Help Pro Expert Videos About wikiHow Pro Upgrade
QUIZZES All Quizzes Hot Love Quizzes Personality Quizzes Trivia Quizzes Taylor Swift Quizzes
EDIT Edit this Article
EXPLORE Tech Help ProAbout UsRandom ArticleQuizzes Request a New ArticleCommunity DashboardTrendingForums Arts and EntertainmentArtworkBooksMoviesComputers and ElectronicsComputersPhone SkillsTechnology HacksHealthMen's HealthMental HealthWomen's HealthRelationshipsDatingLoveRelationship Issues Hobbies and CraftsCraftsDrawingGamesEducation & CommunicationCommunication SkillsPersonal DevelopmentStudyingPersonal Care and StyleFashionHair CarePersonal Hygiene Quizzes Love Quizzes Personality Quizzes Fun Games Arts and EntertainmentFinance and BusinessHome and GardenRelationship Quizzes Cars & Other VehiclesFood and EntertainingPersonal Care and StyleSports and Fitness Computers and ElectronicsHealthPets and AnimalsTravel Education & CommunicationHobbies and CraftsPhilosophy and ReligionWork World Family LifeHolidays and TraditionsRelationshipsYouth
LOG IN
Log in
Social login does not work in incognito and private browsers. Please log in with your username or email to continue. FacebookGooglewikiHow Account No account yet? Create an account
RANDOM
Home
Random
Browse Articles
TrendingNew
Quizzes & Games
All QuizzesHot
Love Quizzes
Personality Quizzes
Fun Games
Dating Simulator
Learn Something New
Forums
Courses
Happiness Hub
Explore More
Support wikiHow
About wikiHow
Log in / Sign up
Terms of Use wikiHow is where trusted research and expert knowledge come together. Learn why people trust wikiHow
Categories
Education and Communications
Studying
Mathematics
Trigonometry
How to Find Coterminal Angles Download ArticleExplore this Article methods 1 Finding a Coterminal Angle 2 Multiple, Least Positive, and Most Negative Coterminal Angles Other Sections Related Articles References Co-authored by Mario Banuelos, PhD and Krysten Jackson
Last Updated: June 3, 2025 References
Download Article X
This article was co-authored by Mario Banuelos, PhD and by wikiHow staff writer, Krysten Jackson. Mario Banuelos is an Associate Professor of Mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels. This article has been viewed 25,716 times.
Coterminal angles are angles that share the same terminal side, the location where an angle stops opening, when drawn in the standard position.[1]XResearch source Finding coterminal angles may sound tricky at first, but the formula is actually very simple once you get the hang of it. Keep reading to learn how to solve this problem.
Things You Should Know
Coterminal angles are angles that stop opening at the same place when drawn in the standard position.
Find any coterminal angle by adding or subtracting 360° or 2π radians from the original angle.
Solve for more than one coterminal angle by adding or subtracting a full revolution multiple times.
Find the most negative and least positive coterminal angles by adding and subtracting until you first cross 0 degrees or radians.
Steps
Method 1 Method 1 of 2:
Finding a Coterminal Angle
Download Article
1 Find your original angle. The angle given to you is the starting point for this problem. The degree of the angle may be provided in the problem or you may need to find it with a protractor or a ruler.
This angle should be in the standard position. The vertex is fixed to the origin of the graph and the initial side, where the angle starts opening, runs along the x-axis.[2]XResearch source
2 Add or subtract 360° when working with degrees. To find a coterminal angle, you must rotate the terminal side in a complete circle. Simply take your original angle and add or subtract 360°.
The formula can be written as θ±360°, where θ is your original angle.
For example, if your original angle was 30°, you may write 30° + 360°. The resulting coterminal angle would then be 390°, or 13π/6 rad if you need to convert to radians.
Adding one revolution would be considered the smallest positive coterminal angle.[3]XResearch source This also applies to radians.
Advertisement
3 Add or subtract 2π rad when working with radians. The formula for finding a coterminal angle in radians is θ±2π, where θ is your original angle and 2π is a complete circle. Simply plug in your original angle to solve.[4]XResearch source
If your θ is π/6 rad, you may set up the problem as π6 - 2π. The resulting coterminal for this equation is -11π/6 rad, or -330° if you need to convert to degrees.
Subtracting one revolution would be considered the smallest negative coterminal angle.[5]XResearch source This applies when working with degrees, as well.
Advertisement Method 2 Method 2 of 2:
Multiple, Least Positive, and Most Negative Coterminal Angles
Download Article
1 Add 360° or 2π radians multiple times to find more coterminal angles. Angles have an infinite amount of coterminals, as you can keep adding or subtracting 360° or 2π rad. If you need to find more than one coterminal angle, just repeat the formula to get more answers.
This formula can be written as θ+360x and θ+2πx, where θ is your original angle and x is the amount of times you need to rotate.
If your original angle was 52°, adding 360° twice will give you 412° and 772°.
Taking the same angle, 52°, subtracting 360° twice will return -308° and -668°.
You can also add and subtract from the same angle to get more than one coterminal. This works great if you need to find both a positive and a negative coterminal angle.
For instance, if you need to find a positive and negative coterminal of π/4, adding 2π will give you the positive result 9π/4 rad and subtracting will give you the negative -7π/4 rad.
2 For "least positive" coterminal, find the last coterminal greater than 0. For a coterminal angle to be the least positive it has to be the last coterminal greater than 0 degrees or radians. To find this, simply subtract 360° or 2π rad until just before you hit the negatives.[6]XResearch source
If your original angle is 361°, the least positive coterminal angle will be 1°. Subtracting anymore will result in negative angles.
Alternatively, the “least positive” will be the first coterminal greater than 0 if your original angle is negative.
Your original angle could be -250°. The least positive coterminal would then be 110°, which is found by adding one revolution.
3 For the “most negative” coterminal, find the first coterminal less than 0. The most negative coterminal angle is the first coterminal angle you find that’s less than 0 degrees or radians. Find this by subtracting 360° or 2π rad from a positive angle until you get your first negative result.[7]XResearch source
For the starting angle 3π/4 rad, the most negative coterminal angle would be -5π/4 rad. This is found by subtracting 2π rad once, which gives a negative angle.
If your starting angle is already negative, the last negative coterminal before your cross 0 would be the most negative.
Let’s say your original angle is -17π/4 rad. The most negative coterminal would be -π/4 rad, which is found by adding 2π twice. Adding another 2π would push you into the positives.
Advertisement
Expert Q&A
Search Add New Question Ask a Question 200 characters left Include your email address to get a message when this question is answered. Submit Advertisement
Tips
Submit a Tip All tip submissions are carefully reviewed before being published Name Please provide your name and last initial Submit Thanks for submitting a tip for review!
You Might Also Like
Easy Steps to Find the Reference Angle in Degrees & RadiansHow toMemorize the Unit CircleHow to Easily Solve Trigonometric EquationsHow toConvert Degrees to RadiansHow to Memorize Trigonometric FunctionsHow toFind Exact Values for Trigonometric FunctionsHow Do You Find the Period of a Function?How toCalculate AnglesHow toRotate a ShapeHow toConvert Radians to DegreesHow to Find Direction of a VectorHow toMeasure an Angle Without a Protractor2 Different Ways to Find a Circle’s Arc Length How to Name Angles: Review Guide for Labeling, Types of Angles & More Advertisement
Co-authored by: Mario Banuelos, PhD Associate Professor of Mathematics This article was co-authored by Mario Banuelos, PhD and by wikiHow staff writer, Krysten Jackson. Mario Banuelos is an Associate Professor of Mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels. This article has been viewed 25,716 times. 7 votes - 74% Co-authors: 6Updated: June 3, 2025 Views: 25,716 Categories: Trigonometry
Print
Send fan mail to authors
Thanks to all authors for creating a page that has been read 25,716 times.
Did this article help you?
Yes No Advertisement Cookies make wikiHow better. By continuing to use our site, you agree to our cookie policy. Co-authored by: Mario Banuelos, PhD Associate Professor of Mathematics 7 votes - 74% Click a star to vote Co-authors: 6Updated: June 3, 2025 Views: 25,716
Quizzes & Games
How Many Digits of Pi Do I Know QuizTake QuizZodiac Compatibility CalculatorAnalyzeCognitive TestTake QuizAccurate IQ Checker Quiz: How Smart Am I?Take QuizGreek Alphabet QuizTake QuizIQ TestTake Quiz
You Might Also Like
Easy Steps to Find the Reference Angle in Degrees & RadiansHow toMemorize the Unit CircleHow to Easily Solve Trigonometric EquationsHow toConvert Degrees to Radians
Trending Articles
Mental Illness TestType A B C D Personality TestCan We Guess How Tall You Are QuizAm I Autistic QuizDiscover the Secret Meaning of Your First NameHow toPractice Kissing
Watch Articles
Amazing Super Bowl Party Ideas Guests Will Cheer ForHow toDevelop Listening SkillsHow toNot Lose Your GirlfriendHow to Draw an OwlHow toTreat AllergiesHow toIntroduce Friends to Other Friends
Trending Articles
5 Different Types of Butts: Find Your ShapeWhat Would My Nickname Be QuizCan We Guess If You're Single or Taken?Depression Test for TeensSCP Fight Simulator: Who Will Win?How toKiss Passionately
Quizzes & Games
IQ Test For KidsTake QuizWhat Tarot Card Am I QuizTake QuizWhat Age Is My Brain QuizTake QuizAm I Smarter Than a High Schooler QuizTake QuizHow Lucky Am I QuizTake QuizThe Impossible QuizTake Quiz
Categories
Education and Communications
Studying
Mathematics
Trigonometry
wikiHow Newsletter You're all set! Helpful how-tos delivered toyour inbox every week! Sign me up! By signing up you are agreeing to receive emails according to our privacy policy.
Download Article Explore this Article methods 1 Finding a Coterminal Angle 2 Multiple, Least Positive, and Most Negative Coterminal Angles Other Sections Related Articles References Co-authored by Mario Banuelos, PhD and Krysten Jackson 
Advertisement 
Co-authored by: Mario Banuelos, PhD Associate Professor of Mathematics This article was co-authored by Mario Banuelos, PhD and by wikiHow staff writer, Krysten Jackson. Mario Banuelos is an Associate Professor of Mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels. This article has been viewed 25,716 times. 7 votes - 74% Co-authors: 6 Updated: June 3, 2025 Views: 25,716 Categories: Trigonometry 
Take Quiz
Analyze
Take Quiz
Take Quiz
Take Quiz
Take Quiz 
Take Quiz
Take Quiz
Take Quiz
Take Quiz
Take Quiz
Take Quiz 
392 
1 Find your original angle. The angle given to you is the starting point for this problem. The degree of the angle may be provided in the problem or you may need to find it with a protractor or a ruler. 
2 Add or subtract 360° when working with degrees. To find a coterminal angle, you must rotate the terminal side in a complete circle. Simply take your original angle and add or subtract 360°. 
3 Add or subtract 2π rad when working with radians. The formula for finding a coterminal angle in radians is θ±2π, where θ is your original angle and 2π is a complete circle. Simply plug in your original angle to solve.[4] 
1 Add 360° or 2π radians multiple times to find more coterminal angles. Angles have an infinite amount of coterminals, as you can keep adding or subtracting 360° or 2π rad. If you need to find more than one coterminal angle, just repeat the formula to get more answers. 
2 For "least positive" coterminal, find the last coterminal greater than 0. For a coterminal angle to be the least positive it has to be the last coterminal greater than 0 degrees or radians. To find this, simply subtract 360° or 2π rad until just before you hit the negatives.[6] 
3 For the “most negative” coterminal, find the first coterminal less than 0. The most negative coterminal angle is the first coterminal angle you find that’s less than 0 degrees or radians. Find this by subtracting 360° or 2π rad from a positive angle until you get your first negative result.[7] 
Advertisement Cookies make wikiHow better. By continuing to use our site, you agree to our cookie policy.