## problem solving with right triangles.docx - Section 3: Discussing the Inverse Relationships

*problem solving with right triangles.docx*

# Solving Right Triangles

Lesson 2 of 13

## Objective: SWBAT solve problems involving right triangles.

## Big Idea: After reading and drawing diagrams for word problems students use the right triangle ratios to solve the problems.

*45 minutes*

#### Bellwork

*5 min*

The plan for today is for my students to use right triangles to solve the problems from yesterday's lesson. To prepare for this, as students arrive I have a diagram with questions about the sides of a right triangle. The questions are designed to review identifying the different sides of the right triangle.

After 2-3 minutes the class will get together to discuss the questions. When we get to questions 3-5, I have students explain and show how to determine the opposite and adjacent sides.

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#### Trigonometry Ratios

*20 min*

After reviewing terminology for describing right triangles, students are assigned to read a portion of page 299 and Example 7 from page 304 of Larson's, Precalculus with Limits, 2nd edition. As students read, they should take notes to prepare for problem solving. They can also discuss questions as a group. If a group has a question that cannot be answered, they are to put the question on a white board for me to see as I move around the room. Eventually, we will discuss any remaining questions as a class.

After reading and note taking, students get out yesterday's problem solving with right triangles worksheet and solve the problems that were diagrammed. Students can continue to add questions to the white board as they work. From time to time, I will stop and ask if anyone can answer the question. If not, we will discuss it as a class.

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In addition to addressing some of the common issues, I ask students new information was contained in the definitions. The new content in the reading were the inverse functions: secant, cosecant and cotangent. Many of my students have little or no memory of these functions when we first encounter them. I will ask them, "Do you notice anything about these new functions and the old functions?"

In my students' minds, they consider these functions to be opposites of the Sine, Cosine and Tangent functions. If this idea is discussed I will say, "So you mean when one is positive, the other is negative?" An intermediate step in the conversation is often the idea the that the function is "flipped". I like to ask if there is a more precise mathematical term for "flipping", and usually someone will finally say reciprocal.

With this terminology in front of the class, we will start to review some of the answers from the worksheet. When we get to Question 3, I expect some groups are still unsure how to find the measure of the angle. Even though we will not discuss the inverse trigonometric functions in detail, today, students can still use the notation to find the angle. So, I plan to work Problem 3 on the board. As I do I will ask several questions:

- How do you undo a function?
- How do we write an inverse?
- What is the function in this problem?
- How can we write the inverse of cosine?
- Can we find this notation on our calculator?

By using the concept of inverse functions here, I want my students to realize that the calculator key cos^(-1) is not the secant key. If a students remarks that they thought cos^(-1) meant secant I remind them that secant is the reciprocal not the inverse. I'll ask, "How do we write a reciprocal?" Generally my students remember that you should write 1/cos x.

After working on Question 3, I ask student if there are any other problems that we need to share. If so, students put the problems on the board. Otherwise the students continue to work on the worksheet.

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#### Closure

*5 min*

As class ends, I give students a closure problem to solve. Students turn this in as an exit slip. The question will let me assess the students understanding of the trigonometric relationships.

As I review the students exit slips I will be assessing whether the students could label the diagram correctly when given a function and its ratio. I will also assess students ability to find the third side.

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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: Review of Angles
- LESSON 2: Solving Right Triangles
- LESSON 3: Law of Sines Introduction
- LESSON 4: Ambiguous Case Day 1 of 2
- LESSON 5: Ambiguous Case Day 2 of 2
- LESSON 6: Finding the Second Solution
- LESSON 7: Problem Solving with Triangles
- LESSON 8: Law of Cosines Day 1 of 2
- LESSON 9: Law of Cosines Day 2 of 2
- LESSON 10: Area of Triangles
- LESSON 11: Review of Solving Triangles Day 1 of 2
- LESSON 12: Review of Solving Triangles Day 2 of 2
- LESSON 13: Solving Triangles Assessment