  # Right Triangle Trigonometry Module

## A practical approach to relating trigonometry to students’ lives.

### Learning Module

• Creator(s):

#### Abstract

This learning module incorporates lessons of right triangle trigonometry in ways that relate the concepts to students’ lives outside of the classroom. Within the module, students are engaged in peer discussion and application of the computational aspects of trigonometry. This lesson fits the Common Core State Standards for Geometry.

# Overview

This Learning Module was designed to help students learn the mathematical concept of trigonometry related to right triangles. Right triangle trigonometry is typically covered in Geometry courses at the high school level. Students taking Geometry are normally in 10th grade. This learning module will incorporate videos from Khan Academy, the use of IXL, as well as online student led activities through Desmos. Ideally, this classroom is a hybrid, where students are able to receive direct instruction and support from the teacher, but also have access to technology to supplement and enhance the learning of the concept. Incorporating the technology approach also allows students to be more independent with their learning of trigonometry. Typically, this unit of learning takes approximately three weeks worth of 45-minute class periods or one and a half weeks of block periods (85-minute).

The learning goals for this learning module are the following. The students will be able to:

• Solve for missing angle measures using properties of right triangle trigonometry.
• Solve for missing side measures using properties of right triangle trigonometry.
• Solve a given right triangle completely using properties of right triangle trigonometry.
• Solve right triangle trigonometry word problems.
• Identify and apply ways in which right triangle trigonometry is used in and relevant to the students' daily lives.

The Common Core State Standards that align with these learning goals are:

• HSG.SRT.C.6 - Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
• HSG.SRT.C.7 - Explain and use the relationship between the sine and cosine of complementary angles.
• HSG.SRT.C.8 - Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
• HSG.SRT.D.11 - Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles.

# 1. Introduction - Prior Knowledge

## For the Student

Prior to studying right triangle trigonometry, students need to be able to:

• Correctly identify and label the parts of a right triangle (legs, hypotenuse, right angle, opposite sides/angles, adjacent sides/angles)
• Understand and apply the Pythagorean Theorem

Assignment: Individually, follow the hyperlink to I and complete Q.1 - Pythagorean Theorem.

- If you miss a question on IXL, use the step-by-step instructions to help you. An example of what this will look like on your screen is shown below (Image retrieved from IXL.com): ## For the Teacher

Purpose: In order for students to be successful with trigonometry, they first need to be able to use and solve problems involving the Pythagorean Theorem. If they are not proficient with this, they will quickly fall behind for the learning objectives involved in this module. This section should be used as a pre-assessment or warm-up to understand the students' level of understanding thus far.

Method: The students will complete section Q.1 Pythagorean Theorem at IXL.com under the Geometry tab. Students will complete this on their own device (either a tablet or laptop). IXL provides instant feedback and step-by-step instructions on how to correctly solve if and when students miss the problem posed. Because of this, the teacher should monitor student progress through the teacher mode of IXL and assist students that are struggling as needed. From this feedback on student progress, the teacher should make small groups for the students. For example, pair students that are not as strong in this topic with students that have mastered it to help foster their growth.

Tips: If you notice one student is quickly finished with the IXL task, have them partnered with another student that may be struggling with this concept. This way, the student gets the help needed while the other student is solidifying their understanding of the Pythagorean Theorem by helping explain it to another student.

# 2. What is Trigonometry?

## For the Student

What exactly is trigonometry??

Suppose you're the head architect for a new skyscraper. Your boss tells you that he doesn't want people craning their necks to see the top of his new building, so the angle of elevation can only be 60º when people are 50 meters away. How tall does the building need to be to fit this criteria? You don't know the distance of the hypotenuse, so using the Pythagorean Theorem is out of the question. This is where trigonometry comes to save the day! View the videos below and take notes in your notebook on the basics of trigonometry. (Videos retrieved from Study.com)  Assignments:

Comment: What is trigonometry? In your own words, summarize what trigonometry is based upon the videos. How is it different from being able to use the Pythagorean Theorem? Give an example of a math problem where you can use trigonometry (and not the Pythagorean Theorem!) to solve it.

## For the Teacher

Purpose: To allow students to get a general understanding of what trigonometry is, how it differs from problems solvable by the Pythagorean Theorem, and how it can be used in an application setting. The comments student make are to get them communicating their own interpretations of what trigonometry is. By sharing their ideas, students will see different explanations from their peers that could help them better understand their own thoughts.

Method: Introduce the construction problem with the whole class first. Hold a brief whole class discussion on how to solve the problem. Project the image of the building problem for students to visualize. Questions to ask: Can we use the Pythagorean Theorem, why or why not? What pieces of information do we need in order to use the Pythagorean Theorem? And is measuring the missing piece a legitimate option to complete in the real world situation? Be sure to close the discussion by stating the Pythagorean Theorem is definitely not an option, so we must find a new one. Use this to transition to Trigonometry. Have the students independently watch both videos and take notes from them to form their own definitions and understanding of trigonometry.

Tips: Once the students have finished watching the videos, allow them to collaborate through their comments. Try not to give them any hints or help on this. It should not be biased by what we as teachers already know about trigonometry.

# 3. SOH CAH TOA

## For the Student

To-Do List for SOH-CAH-TOA:

1. Watch and take notes on the Khan Academy video below on solving for missing side lengths.
2. Follow the link to the Khan Academy lesson on solving for missing angles and using inverse trigonometry.
3. Regroup as a class to discuss and complete the notes for today's lesson. Found here.
Media embedded September 29, 2019

Click here to learn how to use trigonometry to solve for a missing angle.

*pause - redirect your focus for a whole class discussion on our trigonometry notes*

Assignment:

## For the Teacher

Purpose: Allow students to use the resources available to them to figure out the correct way to solve a trigonometric problem. By letting students struggle (a little) to come up with their own solutions, they will retain this knowledge further on.

Method: Instruct students to watch and take notes on the Khan Academy videos. Then pass out their note pages. Complete the first page as a whole class. Then, have the students figure out the method to solve the remaining examples through their own conclusions first. This will help them solidify the concept as opposed to you simply giving them answers. Complete the note pages with the students as a guided practice. However, place it on them to do the work, do not simply give them answers to copy down. Give them time to productively struggle on the examples in their small groups. Provide support as needed, but encourage students to collaborate and use the resources (videos, PDF's of worked out examples) before turning to you as the solution. Give students the opportunity to share their solutions and methods with the entire class. Allow them to be the teacher for this concept and have them verbally walk through and explain their solutions.

A copy of the completed notes is provided in the PDF below.

Tips: If and when students become frustrated with the struggle, lead them in the right direction without giving them the step-by-step instructions. Ask prompting questions that will tap into their prior knowledge like: what information do we need to use sine, cosine, or tangent? what information does the problem already provide us?

# 4. Law of Sines

## For the Student

I know what you're thinking, what happens if there's a problem that could use trigonometry, but it isn't a right triangle? Are we stuck forever knowing we cannot solve the problem?

In your small groups, read through and explore the proof for the Law of Sines below. It appears, we cannot escape using trigonometry even when the triangle is not a right triangle.

In the video from Khan Academy below, work in your small groups to solve the triangle in your notes. Pause the video, solve the triangle using Law of Sines, and then continue the video to check your group's work.

Media embedded September 29, 2019

Assignment:

Create an Update - If there is a Law of Sines, is there a Law of Cosines? How about a Law of Tangents? Use what you learned about in this module to explore these ideas mathematically. Try to discover if there exists a Law of Cosines or a Law of Tangents. Then, screenshot and share your findings in the update.

## For the Teacher

Purpose: The students up until this point think trigonometry can only be used on right triangles. Having them work through the proof of the Law of Sines shows them there are more possibilities. Thus, leading them to exploring into the Law of Cosines as well.

Method: Allow small groups to read through the proof of the Law of Sines provided. Be walking amongst the student groups to give support and extra explanations for this proof as the students need clarification. As the students work through the video example, make sure they are not simply copying down the work, but trying to solve it themselves first. The goal here is for the students to collaborate and come up with correct methods to solve first.

Tips: Since the students are working in groups, take time to check in with students one-on-one for short comprehension checks. There are often students at this point that are not fully engaged in Law of Sines and will simply "follow along" with their group instead of learning it for themselves.

# 5. Law of Cosines

## For the Student

Using what you know about the Pythagorean Theorem, Sine, Cosine, Tangent, and the Law of Sines try to find a new way to solve for the missing side length in the image below: Collaborate within your small groups and work through this example in your notes based on what you've learned about trigonometry thus far.

Once you and your group have come to a common conclusion for this example, watch the video solution below. Be sure to correct your notebook work as needed.

Media embedded September 29, 2019

Assignments:

1. ​​Comment on your Update from the last module. What were you correct about? Modify your update through this comment based on your new knowledge from this module.
2. Complete IXL assignments below:

## For the Teacher

Purpose: This module is to clarify any confusion from the previous module on Law of Sines where students explore the same concept for cosine and tangent. Students should be able to understand these two Laws and know why this is not possible or useful for the tangent ratio by the end of the module.

Method: Allow students to work in their small groups to productively struggle in solving the inital problem solved. Be walking amongst student groups to ensure all group members are participating and engaged in the discourse. Provide assistance in the form of small hints. Remember, the goal should be for the students to discover the formula on their own terms; not simply given it to memorize. This being said, make sure all student groups are grasping the concept. There should be a productive struggle, not just a struggle. As students complete the IXL assignment, monitor their comprehension levels through the teacher interface. An example of what this looks like live is shown below: In this mode, you are able to see, specifically, which students need your assistance. This is also a great way to partner students together. For example, partnering a student that has mastered the skill with one who needs clarification would benefit both students.

Tips: Take advantage of the IXL Real-Time Center to help students as they need it. This is a great way to provide individualized assistance without every student knowing which student (or students) is struggling with the concept.

# 6. Solving Applications of Trig

## For the Student

Despite what you may think, trigonometry does exist outside of math class! Follow the link here to take notes in your notebook on Trigonometri Applications. Collaborate in your small groups while working through the website.

In your small groups, work through the example show in the video from Khan Academy below. Pause the video, work out the problem together in your small groups, then continue the video to check your work.

Media embedded September 29, 2019

Here are a few examples of jobs that use trigonometry:

• Architecture and Engineering
• Music Theory and Production
• Electrical Engineers
• Manufacturing Industry
• Construction
• Cartologists
• Crime Scene Investigators
• Digital Imaging
• Oil Drilling
• Pilots
• Timber Industry

Assignments:

1. Create an Update: Research jobs that use and apply trigonometry. Are there other applications of trigonometry that we haven't discussed yet? If so, give a detailed example of them and how you can use trigonometry to solve the problem posed.
2. Comment: On at least 3 of your peers' Updates, start a conversation by leaving a though-provoking comment relating to thier Update post.
3. Practice trigonometry word problems with the PDF attached below:

## For the Teacher

Purpose: Now that students understand the mathematical procedure of trigonometry, it is vital for students to see and understand how this concept applies to their lives outside of the classroom. This module's focus is to help students solve application problems involving trigonometry.

Method: Monitor student groups as they work through the website. Be looking for misconceptions related to angles of elevation and depression. Students often mix these up and struggle to correctly draw them in their diagrams. Just as before, provide helpful hints and assistance as needed, but let the students productively struggle in their exploration of this concept. The students should be proficient in the mathematical procedure of solving for missing sides and angles at this point. Their focus should be on how to correctly draw and interpret the image of the situation explained in a word problem. Be looking for incorrect diagrams. Take time at the end of this lesson to explain how trigonometry is used in some of the jobs listed as relevant. Then pose the assignment for students to create an update of their own research relating to jobs using trigonometry. Encourage them to engage in this discourse with their fellow students through their comments as well. The answer key for their practice set of word problems is provided in the PDF below.

Tips: If students are struggling to comprehend how an angle of depression is at the top of a triangle, review the concepts of angles created by two parallel lines being cut by a transversal, specifically alternate interior angles being congruent.

# 7. Project - Trigonometry in Your World

## For the Student

"When am I ever going to use this?" said every math sudent ever. You are going to answer this question for trigonometry! In the last module, we started discussing the different jobs relating to and using concpets of trigonometry. Your task is the following:

1. Research a job or real-world situation where using trigonometry is relevant and useful.
2. Create your own trigonometry problem using the research you found. For example, if your research is about an architect using trigonometry, create a trigonometry problem that an architect would need to solve on the job.
3. Solve the problem you create! Once you create your problem, use trigonometry to correctly solve it.
4. Peer review 2 other students' projects.

The detailed PDF of this project is embedded below:

## For the Teacher

Purpose: This module's goal is to allow students to be creative and explore how trigonometry is relevant to their lives. This project allows students to research topics that interest them and make connections to how it relates to trigonometry. By successfully completing this project, the students are showing mastery of the concept by applying and creating thier own trigonometric problem.

Method: Take time to explicitly explain the expectations of the project. Go over the grading procedure of the rubric and ensure that every student has a copy to reference as they work as well. As the students are researching topics, walk amongst them and check in with students' topics to make sure they are relevant and appropriate for the project. Give students plenty of time to fully peer edit each others' work and time to revise their own after receiving peer feedback. The rubric for the project is shown below: Tips: Allow students to be creative with this project, but monitor progress to ensure they are staying focused on the task at hand and not straying from the learning objective. As students are making their own trigonometry problem, pose questions like: what do we need to include in a trigonometry problem? what pieces of information do they give us and which do they expect us to find mathematically? Help students, but do not create their projects for them.

# 8. Student Survey

## For the Student

Please, complete the survey by clicking here.

## For the Teacher

This survey can be used as both a pre- and post-assessment. Its purpose is to collect data on what students comprehend and are still unsure about in relation to the topics covered in this module, based on self-reflection. From the results, you can determine which instructional methods were successful, and which to modify for future use.

Use the data from the survey to modify future lessons as needed. Find areas of common misconceptions and be proactive in preparing for them for future lessons. In the same sense, find areas where students were not challenged enough and supplement those modules to challenge and enhance student learning.

A link of the survey can be found here.

# References

Ferrao, L. "Real life applications of trigonometry." 2018. Embibe. Retrieved from: https://www.embibe.com/exams/real-life-applications-of-trigonometry/

"Solving for a side with the law of cosines." Khan Academy, https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-law-of-cosines/v/law-of-cosines-example

"Solving for a side in right triangles with trigonometry." Khan Academy, https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-solve-for-a-side/v/example-trig-to-solve-the-sides-and-angles-of-a-right-triangle

"Solving for a side with the law of sines." Khan Academy, https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-law-of-sines/v/law-of-sines