Today we formalize the ideas and discuss questions 3 and 4 from Computing Sines and Cosines by Using a Circle. The purpose of the Bell Work is to remind students of what we started yesterday. As we begin the lesson my students have 3-4 minutes to complete the Bell Work problem. The second page of the resource shows how some of my students explained their thinking (students are explaining how they determined the value of sine and cosine in yesterday's activity).
I now want to review Computing Sines and Cosines Using a Circle from yesterday. I begin the review by asking students to share their answer in Question 2. I make sure students see that the hypotenuse of the triangle is the radius of the circle and that the opposite leg is the y while the adjacent leg is the x. This will help students as they discuss Question 3.
I now move to question 3:
In question 3 the term unit circle is introduced, what is meant by a unit circle?
Some students will understand that this is a circle with a radius of 1. If so, I will ask for justification for this choice, "Why is a Unit Circle an productive choice? What does this enable?" If no one volunteers a circle with a radius of 1, then I will explain what this means, then discuss why this might be a choice that can be helpful in a lot of different problems.
For Question 3 I have my students share their answers for Parts a-d. I will ask students to explain how they found the answers. To assist them, I display a unit circle on the board. I ask to students identify the ordered pairs for the points on the x and y axis. Then we discuss whether or not it is necessary to draw a triangle. I'll ask "What did we say the hypotenuse was for the triangle? So do we need to show the triangle?" Once students make a firm connection between the hypotenuse and the radius, then I know that they are understanding how to find the values of sine and cosine on a coordinate plane. It will take some students longer than others to make this connection.
I think Questions 4 and 5 can lead to some great conversations and a deeper understanding of the trigonometric functions. I put both diagrams from Question 4 on the board and let students show how they estimated the values of the trigonometric functions. When we move to Question 5, I will be listening for students to say how the y is larger so sine is larger (or vice versa). I hope that they are beginning to identify patterns that can help them to make estimates that inform their calculations.
As we move through the last two questions, I will make note of how many students are struggling with estimation. I plan to use the last page of the resource as extra problems for groups that need more practice.
I now look at the angles that were more than 90 degrees. I'll say, "When we learned about reference angles I stated that they would be important to find the trigonometric values of an angle." Then I will have my students look at Problem 1b. As they think about the problem I will ask the following questions:
My goal is to clarify for my students why we use reference angles. I want my students to understand how the (x,y) point can be the same for the reference angle, and, the original angle. Seeing this relationship deepens students understanding of how to evaluate real valued functions.
To conclude, we'll refer back to special angles. I will ask my students to find sin 150 degrees. I will give students some time to work on the problem, and then I will have a student share their answer on the board. Afterward, I will give my students other special angles calculations so they can become more fluent using reference angles for commonly used calculations.
As class ends I give the students a prompt to consider:
If you have a point (x,y) on the terminal side of an angle is standard position, how can you find all six trigonometric functions? Write a formula for each function.
Use the formula to find the value of the six trigonometric functions if (-12, -5) a point on the terminal side of an angle in standard position.
Once we read these out loud as a class, I will ask my students to work in pairs and turn in their work as an Exit Slip.