##
* *Reflection: Developing a Conceptual Understanding
Sine and Cosine Day 1 of 2 - Section 2: Finding Sine and Cosine on Circles in the Coordinate Plane

Most trigonometry books have students using the unit circle to find trigonometric values and then transfer to all real numbers. When I used this sequence, I found that students struggled and made mistakes in finding the trigonometric values when given a point such as (3,4). Student would not find the value of 4 since they had in their mind that sin t=y and cos t=x. Once I change to to all real numbers and all points in the plane first the students had less confusion about how to find sine and cosine.

The original activity worked with the unit circle and this still confused students. Again once I changed the activity to be more general my students had less confusion. As we move on in the unit we do the unit circle. The students now do not just think the sin t=y but will always say sin t= y/r and on the unit circle the r=1 so sin t=y.

I also feel that doing the activity in this format helps students understand why reference angles can be used to find the trigonometric values of an angle greater than 90 degrees. Students see how the point on the terminal side give the values for the legs of the triangle.

As we know Common Core asks us to help students develop a deeper understanding of mathematical ideas and I find that today's activity helps my students understand at a deeper level.

*Why not the Unit Circle First*

*Developing a Conceptual Understanding: Why not the Unit Circle First*

# Sine and Cosine Day 1 of 2

Lesson 5 of 13

## Objective: SWBAT evaluate sine and cosine measurements of angles graphed on a coordinate plane.

#### Bell Work

*3 min*

After working with the radian measure of angles for a couple of days, I am ready to introduce my students to trigonometric functions. To recruit my students prior knowledge of working with Sine and Cosine, when students arrive I have the following question on the board:

**How do we use sine and cosine to determine measurements of side lengths and angles when we are given a right triangle?**

Most of my students quickly remember their work in Geometry. For those who do not, this is an opportunity for a quick reminder, with information shared by their classmates, of what they should recall and be able to begin with as we start this topic.

*expand content*

- A handout with several coordinate axes
- A metric ruler
- Protractor
- Compass

To begin, I give students are given the first 3 pages of Computing Sines and Cosines by Using a Circle handout. I save the last (homework) page to see if I will need to use the activity during the lesson. If students meet today's goal of knowing how to find sine and cosine for points in the coordinate plane, than I will not assign the homework.

Students work in **collaborative groups** to do complete the activity. I want students to be able to compare results and help each other as they work. This is the type of activity where one student can dominate by reasoning more quickly than others. I am prepared to move quickly when I sense this is occurring. I will redirect the group by saying, "Each of you need to do the activity. You may compare your results when you are done. The idea is for you to verify your results, not work as quickly as possible. I want your collaboration to increase your accuracy."

I am also looking for mistakes students may make if they do not read the directions thoroughly. They need to follow the instructions and keep clear in their minds which is the initial side and which is the terminal side of the angle. If they do not, they will likely label the point where the circle intersects the x-axis (the point on the initial side) instead of the point where the terminal side of the angle meets the circle. I watch for this issue as students work through the activity. When I sense this is happening, I will correct the problem by asking students identify the terminal side of the angle. (If students are really confused, I will re-teach this topic. The activity will not work if students are not on point.)

If things are going smoothly, after about five minutes I will bring the class together and ask students to share values for** sin 40 **and **cos 40**. I will also ask a couple of students to explain the process that they used to find the values. I want students to hear these explanations from each other.

When students return to working on the activity, I will move around the room helping students when necessary. I expect some students will struggle with the obtuse angles (Problems 1b and 1c). Angles over 180 degrees may also be more difficult. I ask questions such as:

- How can you find the distance the point is along the x-axis? the y axis?
- Will the values be positive or negative? If negative why are they negative?
- How can we use the axis to help us find the terminal side of the angle?

In tomorrow's lesson we will discuss how the coordinates for the angles greater than 90 degrees are the same as they are for the reference angle. This will help students see how to use reference angles in finding trigonometric values.

*expand content*

#### Closure

*10 min*

As today's lesson comes to an end, I bring the class back together for one last thought. This will be an Exit Slip as students leave.

**Let's say the point (3, 4) is on the terminal side of an angle t, which is in standard position. Find the value of sin t and cos t."**

This questions helps me evaluate whether the students understand the trigonometric values can be found if you know a point (x, y) and the distance from the point to the origin, r.

Tomorrow we will formalize this idea. I expect students to complete the first page of activities for the Computing Sines and Cosines worksheets. I assign students to finish questions 4 and 5 at home. Since most students do not have the materials need to do the problems on the homework page I will use that as a reteaching tool. and clarification for tomorrow.

*expand content*

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- UNIT 1: Introduction to Learning Mathematics
- UNIT 2: Functions and Piecewise Functions
- UNIT 3: Exponential and Logarithmic functions
- UNIT 4: Matrices
- UNIT 5: Conics
- UNIT 6: Solving Problems Involving Triangles
- UNIT 7: Trigonometry as a Real-Valued Functions
- UNIT 8: Graphing Trigonometric Functions
- UNIT 9: Trigonometric Identities
- UNIT 10: Solving Equations
- UNIT 11: Vectors and Complex Numbers
- UNIT 12: Parametric and Polar graphs and equations

- LESSON 1: Is John Guilty
- LESSON 2: Radian Measure Day 1 of 2
- LESSON 3: Radian measure Day 2 of 2
- LESSON 4: Coterminal and Reference Angles
- LESSON 5: Sine and Cosine Day 1 of 2
- LESSON 6: Sine and Cosine Day 2 of 2
- LESSON 7: Developing the Unit Circle
- LESSON 8: Evaluating Trigonometric Functions
- LESSON 9: Finding the angle when given the function value
- LESSON 10: How do you find the Inverse of a Trigonometric Functions
- LESSON 11: Using Inverses to Evaluate
- LESSON 12: Review of Trigonometric Functions as Real Valued Functions
- LESSON 13: Assessment of Trigonometric Functions as Real Valued Functions