The Unit Circle (Lesson 9.6)
Use special right triangles to find coordinates on a unit circle.
Understand that on a unit circle, cos = x-coordinate and sin = y-coordinate.
Evaluate sin, cos, tan functions using the unit circle.
This is one of my all-time favorite lessons. Not because it’s super exciting and engaging and students think algebra is so fun, but because it connects right triangle trig to unit circle trig in such a simple way. I don’t know if you know this or not, BUT THE UNIT CIRCLE IS JUST A WHOLE BUNCH OF SPECIAL RIGHT TRIANGLES!!! This truly blew my mind the first time I realized this. I know, some of you are laughing at me because you knew that all along. But I know some of you just had your mind blown too! If we can get students to think of the unit circle with special right triangles instead of memorizing a whole bunch of crazy gimmicks, I can retire now because I will have nothing greater to accomplish in my teaching career.
Alright, so in order for students to get the most out of this lesson, you’ve got to set it up right. Each student needs this unit circle and set of triangles. It’s important that you use these ones because the hypotenuse of the triangles is equal to the radius of the circle.
Students will start out the lesson by finding sides lengths for a 30-60-90 triangle and 45-45-90 triangle that both have a hypotenuse of 1. Once they’ve done that, they should cut out the set of triangles that you give them and label all sides and all angles on both sides of the paper. Students can then use these triangles to fill in the coordinates for all of the points marked on the unit circle by fitting the angles of the triangle into the reference angle of the circle and then using the side lengths of the triangle to determine the x and y coordinates. I would recommend modeling one of the coordinates using a Think Aloud.
As students are completing their circles, walk around and check their work. Once a student has been checked, they can help check other students too. After the unit circle is complete, they can start investigating the relationship between sine and cosine and the coordinates. You could again model the first one.
This is a long activity. Do your best to keep students moving.
Do you know what angle that is? Do we have any angles in our triangles that could fit there?
Where is the reference angle for this angle?
What do you know about x-coordinates in the second quadrant?
What patterns are you noticing?
The majority of the formalization comes when investigating the relationship between the trig functions and the coordinates and in the QuickNotes. The best thing you can do to help students formalize their understanding is to check in with them often. Ask them to explain their thinking. When they’re stuck, use a Think Aloud to model the thought process they should be having.
Consider having a practice day after this lesson, in which students find trig ratios for a variety of different angles. We don’t recommend having students race to see who can fill in the full unit circle the fastest.