How To Find Exact Values For Trigonometric Functions - WikiHow

Skip to ContentQuizzes

• 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. Facebook Google wikiHow 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 Exact Values for Trigonometric Functions Download Article Co-authored by David Jia

Last Updated: February 24, 2025

Download Article

• Learning the Unit Circle
|
• Example 1
|
• Example 2
|
• Q&A

|Show more |Show less X

This article was co-authored by David Jia. David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. This article has been viewed 225,470 times.

The unit circle is an excellent guide for memorizing common trigonometric values. However, there are often angles that are not typically memorized. We will thus need to use trigonometric identities in order to rewrite the expression in terms of angles that we know.

Preliminaries

• In this article, we will be using the following trigonometric identities. Other identities can be found online or in textbooks.
• Summation/difference
  • cos ⁡ ( θ ± ϕ ) = cos ⁡ θ cos ⁡ ϕ ∓ sin ⁡ θ sin ⁡ ϕ {\displaystyle \cos(\theta \pm \phi )=\cos \theta \cos \phi \mp \sin \theta \sin \phi }
  • sin ⁡ ( θ ± ϕ ) = sin ⁡ θ cos ⁡ ϕ ± cos ⁡ θ sin ⁡ ϕ {\displaystyle \sin(\theta \pm \phi )=\sin \theta \cos \phi \pm \cos \theta \sin \phi }
• Half-angle
  • cos θ 2 = ± 1 + cos ⁡ θ 2 {\displaystyle \cos {\frac {\theta }{2}}=\pm {\sqrt {\frac {1+\cos \theta }{2}}}}
  • sin θ 2 = ± 1 − cos ⁡ θ 2 {\displaystyle \sin {\frac {\theta }{2}}=\pm {\sqrt {\frac {1-\cos \theta }{2}}}}

Steps

Part 1 Part 1 of 3:

Learning the Unit Circle

Download Article

1. 1 Review the unit circle.[1] If you are not strong with the unit circle, it is important that you memorize the angles and understand for what quadrants are sine, cosine, and tangent positive and negative.
Advertisement

Part 2 Part 2 of 3:

Example 1

Download Article

1. 1 Evaluate the following. The angle π 12 {\displaystyle {\frac {\pi }{12}}} is not commonly found as an angle to memorize the sine and cosine of on the unit circle.
   • cos π 12 {\displaystyle \cos {\frac {\pi }{12}}}
2. 2 Write the expression in terms of common angles. We know the cosine and sine of common angles like π 3 {\displaystyle {\frac {\pi }{3}}} and π 4 . {\displaystyle {\frac {\pi }{4}}.} It will therefore be easier to deal with such angles.
   • cos π 12 = cos ( π 3 − π 4 ) {\displaystyle \cos {\frac {\pi }{12}}=\cos \left({\frac {\pi }{3}}-{\frac {\pi }{4}}\right)}
3. 3 Use the sum/difference identity to separate the angles.
   • cos ( π 3 − π 4 ) = cos π 3 cos π 4 + sin π 3 sin π 4 {\displaystyle \cos \left({\frac {\pi }{3}}-{\frac {\pi }{4}}\right)=\cos {\frac {\pi }{3}}\cos {\frac {\pi }{4}}+\sin {\frac {\pi }{3}}\sin {\frac {\pi }{4}}}
4. 4 Evaluate and simplify.
   • 1 2 ⋅ 2 2 + 3 2 ⋅ 2 2 = 2 + 6 4 {\displaystyle {\frac {1}{2}}\cdot {\frac {\sqrt {2}}{2}}+{\frac {\sqrt {3}}{2}}\cdot {\frac {\sqrt {2}}{2}}={\frac {{\sqrt {2}}+{\sqrt {6}}}{4}}}
Advertisement

Part 3 Part 3 of 3:

Example 2

Download Article

1. 1 Evaluate the following.
   • sin π 8 {\displaystyle \sin {\frac {\pi }{8}}}
2. 2 Write the expression in terms of common angles. Here, we recognize that π 8 {\displaystyle {\frac {\pi }{8}}} is half of π 4 . {\displaystyle {\frac {\pi }{4}}.} [2]
   • sin π 8 = sin ( 1 2 ⋅ π 4 ) {\displaystyle \sin {\frac {\pi }{8}}=\sin \left({\frac {1}{2}}\cdot {\frac {\pi }{4}}\right)}
3. 3 Use the half-angle identity.[3]
   • sin ( 1 2 ⋅ π 4 ) = ± 1 − cos π 4 2 {\displaystyle \sin \left({\frac {1}{2}}\cdot {\frac {\pi }{4}}\right)=\pm {\sqrt {\frac {1-\cos {\frac {\pi }{4}}}{2}}}}
4. 4 Evaluate and simplify. The plus-minus on the square root allows for ambiguity in terms of which quadrant the angle is in. Since π 8 {\displaystyle {\frac {\pi }{8}}} is in the first quadrant, the sine of that angle must be positive.
   • 1 − cos π 4 2 = 2 − 2 2 {\displaystyle {\sqrt {\frac {1-\cos {\frac {\pi }{4}}}{2}}}={\frac {\sqrt {2-{\sqrt {2}}}}{2}}}
Advertisement

Community Q&A

Search Add New Question

• Question How do I find the exact value of sine 600? Donagan Top Answerer 600° = 60° when considering trig functions. [600 - (3)(180) = 60] Sine 600° = sine 60° = 0.866. Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 70 Helpful 15
• Question What does ASTC stand for in trigonometry? Donagan Top Answerer It stands for the "all sine tangent cosine" rule. It is intended to remind us that all trig ratios are positive in the first quadrant of a graph; only the sine and cosecant are positive in the second quadrant; only the tangent and cotangent are positive in the third quadrant; and only the cosine and secant are positive in the fourth quadrant. Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 8 Helpful 16
• Question What's the exact value of cosecant 135? Donagan Top Answerer You can find exact trig functions by typing in (for example) "cosecant 135 degrees" into any search engine. Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 88 Helpful 10

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

How toMemorize the Unit Circle How to Easily Solve Trigonometric Equations How to Memorize Trigonometric Functions How Do You Find the Period of a Function? Easy Steps to Find the Reference Angle in Degrees & Radians How to Find Coterminal Angles How toFind the Fourier Series of a Function How toUse the Sine Rule How toUse the Cosine Rule How toIntegrate Using the Beta Function 2 Simple Ways to Calculate the Angle Between Two Vectors How toAdd and Subtract Square Roots How toCalculate Probabilities of Quantum States How toIntegrate the Sinc Function Advertisement

References

1. ↑ https://www.mathsisfun.com/geometry/unit-circle.html
2. ↑ http://mathworld.wolfram.com/TrigonometryAnglesPi8.html
3. ↑ https://www.mathway.com/popular-problems/Precalculus/400452

About This Article

Co-authored by: David Jia Math Tutor This article was co-authored by David Jia. David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. This article has been viewed 225,470 times. 111 votes - 45% Co-authors: 3 Updated: February 24, 2025 Views: 225,470 Categories: Trigonometry

• Print
• Send fan mail to authors

Thanks to all authors for creating a page that has been read 225,470 times.

Reader Success Stories

• 

  M L Agrawal

  Nov 10, 2021

  "This article helped me to understand basic principles of the trigonometric table."

More reader stories Hide reader stories Share your story

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: David Jia Math Tutor 111 votes - 45% Click a star to vote 45% of people told us that this article helped them. Co-authors: 3 Updated: February 24, 2025 Views: 225,470

M L Agrawal

Nov 10, 2021

"This article helped me to understand basic principles of the trigonometric table."

Susmit Pathak

Jun 10, 2018

"Very helpful for students like me!" Share yours! More success stories Hide success stories

Quizzes & Games

Greek Alphabet QuizTake QuizHow Many Digits of Pi Do I Know QuizTake QuizIQ TestTake QuizCognitive TestTake QuizMusic Notes & Symbols TestTake QuizWhat Tarot Card Am I QuizTake Quiz

You Might Also Like

How toMemorize the Unit CircleHow to Easily Solve Trigonometric EquationsHow to Memorize Trigonometric FunctionsHow Do You Find the Period of a Function?

Trending Articles

How toLook Your BestWhat Emojis Mean Sex?The Different Kinds of Dimples: Types, Causes, & Social Perception151 of the Juiciest “Most Likely To” Questions to AskSigns a Woman is Sexually Attracted to YouDo You Agree with These Hygiene Hot Takes?

Watch Articles

How toCalculate the Volume of a PyramidThe Best Way to Exfoliate Your Scalp (Plus, What to Use)How toSave Money as a KidHow toPolish Aluminum2 Easy Renter-Friendly Options to Hang Your Window TreatmentsHow to Insert Slide Numbers in PowerPoint

Trending Articles

Why Can't I Sleep QuizHow toBe PrettyThe Most Attractive Zodiac Signs & What Makes Each Sign Beautiful24 Different Types of Bras ExplainedHow to Play the Concentrate Game (For A Little Scare!)What Does Your Rice Purity Score Really Mean?

Quizzes & Games

What Age Is My Brain QuizTake QuizWhat Zodiac Sign Am I QuizTake QuizGeneral Knowledge QuizTake QuizAm I a Genius QuizTake QuizKinsey Scale TestTake QuizLeetspeak Translator & GeneratorGenerate

• 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.

• Home
• About wikiHow
• Experts
• Jobs
• Contact Us
• Site Map
• Terms of Use
• Privacy Policy
• Do Not Sell or Share My Info
• Not Selling Info
• Contribute

Follow Us

×

wikiHow Tech Help Pro:

Level up your tech skills and stay ahead of the curve

Let's go! X --539