##
* *Reflection: Vertical Alignment
Using Trigonometry to Solve Right Triangles - Section 1: Learning the Decision-Making Framework

If I had to pick one concept that is central to high school mathematics, it is the concept of functions. I'll admit that when I went through high school, I had no idea what a function was apart from it having something to do with f(x) and the notation never did sit quite right with me.

I'd have to say that students tend to be just as confused when it comes to functions, particularly trigonometric functions, especially the inverse trig functions. So in this lesson, I really try to make connections to the concept of functions.

For any function, I like for my students to be able to tell me what its "purpose" is. I would want them to know that sine, cosine and tangent are functions that "take" an angle measure as the input and "return" a ratio as the output. Once they understand this, they'll be less likely to think that sin50 means sine times 50, for example.

Next I want them to understand that the inverse of a function takes what would be the output of the function and returns the input that would have produced it. So the inverses of sine, cosine and tangent take ratios as their inputs and return angle measures as their outputs. When students begin to understand this, then they'll be more clear on when they should use sine, for example, and when they should use "sine to the negative first power"...JUST KIDDING.

So again, building students' literacy around functions is a long term process that hopefully is occurring throughout the entire four years of high school.

*Building Knowledge of Functions: Inverses*

*Vertical Alignment: Building Knowledge of Functions: Inverses*

# Using Trigonometry to Solve Right Triangles

Lesson 3 of 6

## Objective: SWBAT use trigonometry to solve for missing side lengths and angle measures in right triangles.

In this section, I teach students the basic thought process and mechanics for solving right triangles using trigonometry. I give each student Solving Right Triangles. This is a teaching worksheet that requires students to do some reading and comprehension.

We read through the packet together. I regulate the pace of the reading so that students are not speeding through without understanding what they are reading. I also rephrase, reiterate what is being said in the text just to reinforce it so that students are understanding. When the handout calls for students to fill in blanks or solve problems, I first ask the students to work independently. Then I have them compare their responses with their A-B partners. Finally, I reveal the answer or demonstrate the solution process.

After the first page of the handout, I want to make sure that students understand when they can and cannot use Pythagorean Theorem and when they can and cannot use Trigonometric ratios to solve for the missing side length in a right triangle.

On page 2, I introduce a decision-making framework that helps students to understand problems and map out a solution path. Students get to solve problems (two ways) on their own on page 3. The method1/method2 format on page 3 allows us to talk about sinx= cos(90-x) and tanx = 1/tan(90-x)

On page 4, I discuss inverse functions. At this point the text gets dense so I usually stop to write on the whiteboard to make sure students are getting the ideas being presented. At the end of this page, students should understand that sine, cosine and tangent are functions that take angle measures and output ratios. They should also know that the inverse trig functions take ratios and output angle measures. They should also know how to use inverse trig functions to find unknown angle measures in right triangles.

#### Resources

*expand content*

#### Practice Problem Solving

*30 min*

At this point, I turn to the textbook to find application problems that involve angle of elevation and depression. I find that it is important to model a good number of these problems for students so that they can see the thought process that goes into solving these problems. I spend 15-20 minutes modeling problems.

I repeatedly use the decision-making framework. Asking myself aloud "Ok...what is my point of view? Which side do I have? Which do I desire? Which ratio should I use? Is my variable in the numerator or in the denominator?

For me, teaching students to organize their brains for decision making is one of my broader goals in the geometry course and this is a good opportunity to make progress on that goal.

One thing I stress in my demonstrations is postponing calculations until the very end. For example if we have sin 34 = x/15, I would caution students against finding sin 34 at this juncture. Instead it would be better to say x equals 15sin34 , and is approximately equal to 8.39.

I give students numerous problems to start in class and finish for homework. During my demonstration, I tell students that they have two options: They can pay attention to my modeling or if they are ready to work independently they can do so quietly without communicating with anyone else. That way students who want to pay attention to the demonstrations can do so without distractions and those who feel confident can get the practice they need.

*expand content*

Some students like a challenge. Not only that, a broader goal in the class is to get students used to the idea that some problems don't have numerical answers. These problems force us to deal with structures and relationships as opposed to focusing on computations. One such problem that will give students a challenge is the Solving Right Triangles Challenge Problem.

I give students a day or two to work on this at home and I demonstrate the solution once students have all had their go at it.

*expand content*

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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