Want to know the best way to teach trig functions in unit circle? This step-by-step guide is exactly what you need.
Knowing the unit circle well is crucial for doing well in upper level mathematics. If you’re a math teacher like me and are having a hard time getting your students to understand how triangles and circles go together, this is the absolute best guide to teaching trig functions in unit circle.
You’re going to learn how to teach the unit circle to your high school math students in a fun creative way, from how to introduce the lesson to how you can assess your students for understanding.
After following this step-by-step guide, you are going to be an expert at having your students make connections between the unit circle and trig functions.
This post is all about teaching trig functions in unit circle.
Best Guide to Teaching Trig Functions In Unit Circle
1. The Introduction: Activate Prior Knowledge
The first thing you need to do at the beginning of your lesson is to activate prior knowledge. Since you’ll be teaching students how to use trig functions in the unit circle, you will need to review the following with them:
b) Special Right Triangles: 45-45-90 and 30-60-90 Triangles
c) SOH-CAH-TOA (the 6 trigonometric ratios)
You can use a variety of activities to activate prior knowledge such as K-W-L or a quick color by code or other activity to get them to remember how to use these concepts to find missing side lengths in a triangle.
Sample Activities you can use:
- Trig Functions for Triangles
- Trigonometry BINGO (Pythagorean Theorem, SOH-CAH-TOA, Sine and Cosine Rule)
- SOH-CAH-TOA Trig Review
- Trigonometry Task Cards
- Trigonometry Review Quiz
2. Define and Graph the Unit Circle on a Rectangular Coordinate Plane
Once you are confident that students can find missing side lengths in a right triangle using the three concepts mentioned in #1 of this guide, you are ready to introduce the concept of the unit circle. You can explain to your students that the unit circle is simply a circle centered at the origin with a radius of 1.
On a coordinate plane, draw a large circle with a radius of 1. Ask students to identify the four intercepts on the unit circle.
Then ask students what is the circumference of a circle? They should answer: it’s the distance around a circle. Recall the formula for the circumference of a circle and so explain how this relates to radian measure.
3. Divide the Unit Circle into 4 Equal Parts
Now have students divide 360 degrees by 4 to determine the four intercepts that are on the circle in degrees.
Do the same now but in radians.
4. Divide the Unit Circle into 8 Equal Parts
Now tell students to divide the circle into 8 equal parts.
5. Derive the coordinates of 45 degrees by using the special right triangle 45-45-90.
Now you can go around the circle and explain why the signs change depending on the quadrant the point is located.
6. On a New Circle, Have Students Divide the Circle into 12 Equal Parts
The same way you were able to split the circle into 8 equal parts, now you’ll show students to split the circle into 12 equal parts. Have them draw a new large circle on their paper.
7. Derive the coordinates of 30 degrees by using the special right triangle 30-60-90.
8. Derive the coordinates of 60 degrees by using the special right triangle 30-60-90.
9. The Unit Circle
You’re ready to show your students the entire unit circle. Now they will have understood how the coordinates were derived. It’ll make it much easier for them to understand the relationship between trig functions and the unit circle.
Explain to your students that just as before, the signs of the coordinates will change depending on the quadrant they’re located in. Anything that is pi over 6 will have the same coordinates (with different signs of course depending on the quadrant) and anything that is pi over 3 will have the same coordinates (again, different signs according to the quadrant they’re located in).
10. The Unit Circle Project
This Unit Circle Project is the perfect way to teach the unit circle to your high school math students in a fun creative way. Not only will students walk away with a better understanding of special right triangles and angle relationships after completing the unit circle project, but you’ll also be able to decorate your classroom with their artistic ideas. Your administrators and parents will love seeing the artistic aspect of this unit circle math project.
Instead of giving students the unit circle to memorize, it’ll be more meaningful for your students to see how the unit circle comes about and why the coordinates are what they are. This unit circle project is meant to help students in both remembering the unit circle and understanding it. There are 2 worksheets included in this project: Unit Circle Project Background Information and The Unit Circle. The first can be done whole class as a way to introduce the lesson/project. The second can be done as part of the project or you can even use it as a QUIZ after students complete the project.
Give each student a Unit Circle Project Directions worksheet. I assign this project independently. It’s quite easy to grade with a print and use rubric that’s included. Since the unit circle rubric is out of 10 possible points, you can easily convert to a percent and assign a letter grade. Ask students to attach the rubric to the back of the project for your convenience. You can even ask them to grade each other’s work.
- Unit Circle Background Information Worksheet with Answer Key
- Unit Circle Practice Worksheet or Quiz (depending on how you use it) with Answer Key – students must find all six trig functions for 8 points on the unit circle
- Unit Circle Project Directions
- Grading Rubric – print ready; 2 per sheet
- Unit Circle Project Examples
Teacher Reviews of The Unit Circle Project:
- Thank you so much for creating this project. My 11th grade pre-Calculus students were having a very hard time with the unit circle. They had no problem understanding the circle, the degrees and the radians. They had no trouble understanding trig principles of SOA CAH TOA but when I taught combining of the two they got very confused how triangles and circle went together. I taught them my way, we watched a Khan Academy video a different way but they still struggled.
This project was the breakthrough. I also have some students who are very artistic, always drawing and making beautiful notes so this gave them a chance to be creative. I would highly recommend this project.
- I really liked implementing this project in my Precalculus class. The students were able to spend time understanding where the angle on the unit circle come from and they really enjoyed being able to show their artistic side. As a bonus, I have very colorful and neat work to show off in my classroom.
- I am so happy I purchased and used this with my Algebra 3 students this year! Algebra 3 is a blend of Alg 2 and Pre-Calculus. We ended the semester with the unit circle and it was a fantastic way to end our time together. I wish I could share some of the products I received from my student.
I was BLOWN AWAY at their creativity and effort they put into this!
This is one of my absolute favorite projects to do with my students, and it’s one of the FIRST products that I ever posted on Teachers Pay Teachers. Unfortunately, there have been several people who have posted the project directions on the internet, even though the the terms of use clearly state: “This resource may not be uploaded to the internet in any form, including classroom/personal websites or network drives. Leave feedback to earn credits for future purchases.” While I have tried to get these violators to take it down, it’s quite the arduous task and requires getting my lawyer involved. I guess the good news is that others do find the project helpful and it is nice to see the creativity of students worldwide in using this project that I came up with years ago when I first started teaching the unit circle to my IB students.
Hope you enjoy it too!
So there you have it, a complete step-by-step breakdown of how you can teach the unit circle to your students and how you can bring together circles and triangles in a way that will be easy for them to understand. The Unit Circle Project is a nice way to bridge this gap and get your students’ creative juices flowing. You won’t regret doing this project with your students.
