The Unit Circle Day 1 of 2

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Students will be able to define trigonometric functions of acute angles

Big Idea

Why do we use a unit circle? How can we use it to represent all angles for all circles?

Warm-up and Homework review

10 minutes

I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative explains this lesson’s Warm Up - Unit Circle Day 1 which asks students to find the sine of an isosceles triangle.


Guided Investigation #1

20 minutes

My goal for this section is the give the students a solid understanding of why we use a unit circle and the fact that the size of the circle doesn't matter when looking at trigonometric ratios. 

We begin by looking at the trig ratios of similar triangles on the coordinate plane.  The students use the Pythagorean Theorem and what they know about trig ratios to find the sine, cosine and tangent each time.  Once most students have the ratios, I ask them to analyze and make a conclusion about the three problems.  I then ask for volunteers or call on someone who had particularly good insights.  The goal is to get one of the students to say that they are similar triangles with the same angles and therefore the ratios are the same.                

The next slide gives a good visual of this fact.  This can be used as an aid when making conclusions about the nature of these three ordered pairs.  A potential extension would be to discuss how all of the hypotenuses are of the same line (Math Practice 7). 

We then look at the Pythagorean Theorem using a 3,4,5 triangle.  I talk about how we can reduce all these circles to a single circle for the sake of simplicity.  This is the Unit Circle, a diagram of which will then come up in proportion to the original circle.  The final step is for the students to figure out what the x and y values will be on the unit circle.  I let the students struggle with this for a bit (Math Practice 1).  As scaffolding support, I may ask the students how to turn 5 into a one.  Some students may want to say the x is 16/25 and y is 9/25.  I then ask them how we deal with the squares.  Depending on the success of the students, either I have a student explain how they found x and y or walk through it together as a class. 

In the concluding activity of the investigation, students connect the fact that the sine and cosine are the same as the y and x values.  It will take a bit more work to show that tangent is sine/cosine so we may do this as a class.  It all depends on the kids. 

We then put everything from the investigation together.  This is the BIG IDEA of the day.  

Guided Investigation #2

15 minutes

This goal of this section is to review the most important angles to trigonometry.  I do not require my students to memorize the trig ratios for these angles but I do want them to know how to produce them from a 45-45-90 and a30-60-90 triangle.  The memorization will come in Pre-Calculus.  They have already looked at these triangles in an earlier lesson.  This lesson will place them on a unit circle.  

First, the students are asked to find the ratios of the three sides on a unit circle and then identify the sine, cosine and tangent.  I have students work on this independently or in pairs.  This will prove problematic for some students as the equation required to find this problem is x2 + x2 = 1.   I walk around and give hints to students who may need it.   

Next the students produce the same thing for a 60o and then a 30o angle (Math Practice 7).  I think it is very important that they associate these two angles with the bisection of an equilateral triangle.  This will aid them in reproducing these triangles to find the trig ratios.    


The first four problems give the students a coordinate on the unit circle and asks them to find the trig ratios off of it. 

The next four problems give the students one of the trig ratios and asks them to find the coordinate on the unit circle that has that ratio.  These where not introduced during the lesson and may provide a challenge to some students (Math Practice 1 and 7).  Not only do students need to understand the structure behind the unit circle and use it to solve a new type of problem, they need to persevere in attempting to figure out how to solve that new problem. 

The final problem asks students to fill in a chart with the trig ratios for sine, cosine and tangent for both 30o and 60o.  They are then asked to talk about anything they notice about this chart (Math Practice 3).  They should notice and talk about  how sine and cosine of one angle is equal to the other opposite of the other. 

Exit Ticket

2 minutes


This Exit Ticket asks the students to explain why we use a unit circle to find trigonometric ratios (Math Practice 3).  It should provide a good indicator as to how well students understood the lesson.