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* *Reflection: Diverse Entry Points
Using the Unit Circle to Prove the Law of Sines for Obtuse Triangles - Section 1: A Broader Conception of Angle Measure

Conceptual change is difficult...so the research says. Anyhow, I remember the first time I saw the standard G.CO 1 which has as one of its components that students will define an angle as a distance along a circular arc. Wow, I thought, what a fundamental change that's going to be from the basic idea of two rays with a common endpoint. How do you just tell someone that the definition of angle is now this? It's like trying to convince people that Pluto is not a planet when it's been a planet their whole lives.

So having some show to go with the tell is a good help. I was really happy and thankful to the creators of Sketchpad when I found a way to represent this definition of angle visually for students. So in this era when we do have the power of technology, I'm always looking for ways to visually represent concepts that are hard to understand or accept.

*Good visuals help with paradigm shifts*

*Diverse Entry Points: Good visuals help with paradigm shifts*

# Using the Unit Circle to Prove the Law of Sines for Obtuse Triangles

Lesson 3 of 5

## Objective: SWBAT connect the trigonometric ratios with the unit circle

Up until this point, my students still have the idea of an angle as two rays with a common vertex. So this is a major conceptual shift. I start by providing some context with physics. I explain how in physics there are two types of motion: linear motion and angular motion. Linear motion, I explain, is motion in straight lines. Angular motion, on the other hand, is motion along a circular arc. So for example when a track athlete is running the 100 meter dash, we say they are running at a velocity of 10 meters per second, for example. But if an ice skater is spinning during a routine, we might say that she is rotating with an angular velocity of 700 degrees per second.

After this brief anecdote, I will give a definition of an angle measure as a distance along a circular arc. To bring this definition to life, I show students a Geometer's Sketchpad sketch that you can see in the following video.

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Earlier in the course, I've taught a lesson on the Law of Sines. In that lesson, we only proved the law for acute triangles because the proof for obtuse triangles relied on students having some knowledge of the unit circle. So at the time, I promised my students that later in the year, I would come back and teach them about the unit circle so that we could prove the law of sines for obtuse triangles. Hence, this lesson.

I start by providing a basic introduction to the unit circle and I develop the identity sin (180-x) = sin x. Finally students prove the law of sines for obtuse angles.

All of that takes place using the Law of Sines for Obtuse Triangles handout as a medium.

I do some direct teaching on the unit circle and trigonometry thereof. I also give a pretty direct explanation of the sin(180-x) = sin x relationship. I just want students to know this information and understand the concepts involved.

Once students get to the last two pages of the handout, they will be working independently to apply what they learned earlier and what they have learned in this section of the lesson.

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- UNIT 1: Community Building, Norms, and Expectations
- UNIT 2: Geometry Foundations
- UNIT 3: Developing Logic and Proof
- UNIT 4: Defining Transformations
- UNIT 5: Quadrilaterals
- UNIT 6: Similarity
- UNIT 7: Right Triangles and Trigonometry
- UNIT 8: Circles
- UNIT 9: Analytic Geometry
- UNIT 10: Areas of Plane Figures
- UNIT 11: Measurement and Dimension
- UNIT 12: Unit Circle Trigonmetry
- UNIT 13: Extras

- LESSON 1: Deriving Formulas for Sector Area and Arc Length
- LESSON 2: Define Radian Measure
- LESSON 3: Using the Unit Circle to Prove the Law of Sines for Obtuse Triangles
- LESSON 4: Converting between Degrees and Radians
- LESSON 5: Introduction to Trigonometric Functions and their Graphs