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# Sine and Cosine on the Unit Circle

Lesson 4 of 9

## Objective: SWBAT use the unit circle to extend the definitions of sine and cosine to all real numbers.

## Big Idea: Now that we've established the structure of the unit circle, it's a short leap to understaning our new, "extended" definitions of sine and cosine.

*72 minutes*

#### Opener - Check in Quiz

*2 min*

As they enter the room, I welcome each student and give them an index card. I say that when the bell rings, they’ll have two minutes to complete today’s Check_In_Quiz. When the bell rings, I put up slide #3 on today’s Prezi. On the slide are learning target Unit Circle 1 (which matches CCS HSF-TF.A.1) and the prompt, “In 10 words or less, explain why the radian measure of an angle is equivalent to the length of the arc is subtends on the unit circle.”

The answer I’m looking for will include the idea that the radius has length 1, and perhaps a mention of the arc length formula. If students nail it, I’m happy to give them a 4 on this SLT. Whether they get it or not, I think of assessments as a chance for students to learn something. I often find that calling something a “quiz” is exactly what makes the knowledge click for some students.

#### Resources

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#### Notes and Guided Practice

*35 min*

Today's lecture notes draw on the work students have done over the last few days. By working to complete Figure 1, the Unit Right Triangles handout, and Figure 2a, students have built background knowledge. Now we will try to synthesize that knowledge and use it to understand the definitions for sine and cosine on the unit circle.

This lecture consists of discussion questions, math problems, and student tasks that we will move through together as a class. Today's** Explaining the Math **resource provides details of how I move through the lecture. I recommend taking a look at today's Prezi to think about how you would use it, then referring to my extended notes in Explaining the Math to learn more about my enactment of this lesson.

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#### Work Time

*30 min*

See the final slide of today's Prezi. Students can use the remaining time however they see fit. Most will either finish Figure 2a by reflecting their triangles on the x-axis, or they will grab a laptop and work on the Delta Math exercises:

Continued, focused work on the Delta Math is central to the next lesson.

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#### Closing + Homework

*5 min*

To close out today's lesson, I collect index cards as evidence of what students plan on doing next. I also assign Problem Set 13 for homework.

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- UNIT 1: Statistics: Data in One Variable
- UNIT 2: Statistics: Bivariate Data
- UNIT 3: Statistics: Modeling With Exponential Functions
- UNIT 4: Statistics: Using Probability to Make Decisions
- UNIT 5: Trigonometry: Triangles
- UNIT 6: Trigonometry: Circles
- UNIT 7: Trigonometry: The Unit Circle
- UNIT 8: Trigonometry: Periodic Functions

- LESSON 1: What's so Great About the Unit Circle?
- LESSON 2: The Unit Radius and the Unit Hypotenuse
- LESSON 3: Using Right Triangles to Make the Unit Circle
- LESSON 4: Sine and Cosine on the Unit Circle
- LESSON 5: Work Period: Extending the Definitions of Trig Ratios to the Unit Circle
- LESSON 6: Unit Circle Problem Solving, Synthesis, and Discovery
- LESSON 7: Tangent on the Unit Circle and Relationships between Trig Functions
- LESSON 8: Work Period: Unit 3 Review
- LESSON 9: Unit 3 Exam