# 3.8: Right Triangle Trigonometry

## The Pythagorean Theorem

I. Section Objectives

- Identify and employ the Pythagorean Theorem when working with right triangles.
- Identify common Pythagorean triples.
- Use the Pythagorean Theorem to find the area of isosceles triangles.
- Use the Pythagorean Theorem to derive the distance formula on a coordinate grid.

II. Cross- curricular-Toy Construction

- If possible, complete this after watching the movie.
- Divide the students into groups of three or four.
- You will need Kynex for this activity.
- Students may be able to bring in some from home.
- If Kynex are not available, just have this be a design project.
- Tell the students that they are going to be designing a toy that has a right triangle as its core component.
- Students can use other shapes as well, but the triangle is central.
- Students are to draw a design of their toy.
- Then, students are to build a model using the Kynex.
- Allow time for students to share their work when finished.

III. Technology Integration

- Use the following website so that students can watch a short movie on creating triangular toys.
- www.thefutureschannel.com/dockets/realworld/inventing_toys/
- This video shows how two designers working for Kynex design toys.
- Tell the students to notice all of the uses of polygons and triangles in the designs.
- When finished, discuss the video.
- What did the students observe?
- What did they notice about the shapes used in the toy designs?
- How did patterns impact the work of the designers?
- How does geometry impact their work?

IV. Notes on Assessment

- Assess each toy design and construction.
- You may want to create a rubric for grading the toys.
- Observe students as they work.
- Provide students with feedback when necessary.

## Converse of the Pythagorean Theorem

I. Section Objectives

- Understand the converse of the Pythagorean Theorem.
- Identify acute triangles from side measures.
- Identify obtuse triangles from side measures.
- Classify triangles in a number of different ways.

II. Cross- curricular-Architecture/Design

- Use the following image from Wikipedia to show students an image of St. Basil’s Cathedral.
- This is Figure 08.02.01
- www.en.wikipedia.org/wiki/File:RedSquare_SaintBasile_(pixinn.net).jpg
- You can either use this image as a discussion point or have students work with it in small groups.
- In small groups, have the students identify the equilateral and acute triangles in the cathedral.
- There are many of them to choose from.
- Then ask the students to identify how they know that these are equilateral and acute.
- The students should be able to discuss the different characteristics of what makes an acute triangle acute and what makes an equilateral triangle equilateral.
- Have students discuss this in small groups.

III. Technology Integration

- Ask students to research triangles and bridge designs.
- What is the most common type of triangle used in bridge designs?
- Why is it the most common?
- Have the students do some research on this and then report on their findings.
- Students should keep track of any websites they visit to refer back to when reporting on their findings.

IV. Notes on Assessment

- Observe students as they work.
- Listen to the discussions and you will hear whether the students have an understanding of acute, obtuse and equilateral triangles.
- Ask questions to expand student thinking.

## Using Similar Right Triangles

I. Section Objectives

- Identify similar triangles inscribed in a larger triangle.
- Evaluate the geometric mean of various objects.
- Identify the length of an altitude using the geometric mean of a separated hypotenuse.
- Identify the length of a leg using the geometric mean of a separated hypotenuse.

II. Cross- curricular-Triangular Lodge

- Have students use this website, or show them the image and give them the measurements that they will need to work with.
- www.daviddarling.info/encyclopedia/T/Triangular_Lodge.html
- This is a building that is composed on a triangle.
- We know that each side of the triangle is long.
- If this is the case, what is the altitude of the building?
- Have student work in small groups or pairs to solve this problem.
- Students will need to work through the formula for geometric mean in the text.
- If they are having trouble, refer them back to the text for this information.
- Solution:
- Be sure that the students understand how the measurements are all the same.
- Allow time for questions and feedback.

III. Technology Integration

- Have students complete some research on circus tents.
- Circus tents use poles and canvas to hold up the tent.
- The use of the poles impacts the height or altitude of the tent.
- Ask the students to report on the most common design of a circus tent.
- Have them make a list of the websites that they visit and to select one type of tent or image to discuss.
- You can conduct a discussion about how geometric mean, altitude and triangles connect with circus tents.
- How are they interconnected?
- This will require the students to use higher level thinking skills since the connections may not be obvious.

IV. Notes on Assessment

- Assess student understanding through discussion.
- Try to have time for each group to share.
- You will see how much the students understand through their sharing and conversation.

## Special Right Triangles

I. Section Objectives

- Identify and use the ratios involved with right isosceles triangles.
- Identify and use the ratios involved with triangles.
- Identify and use ratios involved with equilateral triangles.
- Employ right triangle ratios when solving real- world problems.

II. Cross- curricular-Sports

- Use the following image of a baseball diamond from Wikipedia.
- This is Figure 08.04.01
- www.en.wikipedia.org/wiki/File:Baseball_diamond_marines.jpg
- This is a problem to solve.
- Here is the problem.
- If the distance between the bases is , how far will the first baseman throw the ball to reach the third baseman?
- Solution:
- To solve this problem, you can use the Pythagorean Theorem since each of the bases is at a angle.
- Therefore, you can split up the baseball diamond into triangles.
- is the distance from first to third base.

III. Technology Integration

- Have the students complete a websearch on baseball fields across the United States.
- Students can select their favorite one and report on its dimensions.
- Does the Pythagorean Theorem work for all baseball diamonds?
- Conduct a discussion exploring the angles and dimensions of baseball diamonds.

IV. Notes on Assessment

- Were the students able to solve the problem?
- Were there struggles?
- Did the students see the right angles in the diamond?
- Did they notice that they could divide the diamond into two triangles?
- Where is the hypotenuse of the triangles?
- Assess student work and provide feedback as needed.

## Tangent Ratios

I. Section Objectives

- Identify the different parts of right triangles.
- Identify and use the tangent ratio in a right triangle.
- Identify complementary angles in right triangles.
- Understand tangent ratios in special right triangles.

II. Cross- curricular-Art/Furniture Making

- Have the students look at the website or show them the images of the triangle table.
- You can use this as a discussion piece.
- Ask the students to identify the parts of the right triangle.
- Then ask them to identify the tangent ratio of the right triangle.
- Finally, students can be given the task of constructing their own right triangle table.
- Students will need tools and saws to do this.
- You may want to see if you can combine this activity with woodshop, if offered in your school.
- Have the students share their work when finished.

III. Technology Integration

- Have the students explore the concept of dragon tiles that have right angles in them.
- The students can go to the following website to explore this.
- www.ecademy.agnesscott.edu/~lriddle/ifs/levy/tiling.htm
- This will provide students with step by step directions on how to complete the dragon tiles.
- Have students work in small groups.
- When the students have finished studying the information on the website, have them go ahead and create their own pattern of dragon tiles.
- Students can work in pairs on this.
- They can either draw in each tile, or create a pattern to trace.
- Either way, the dragon tiling will be completely made of right triangles.

IV. Notes on Assessment

- Assessment can be completed by looking at each student’s work product.
- If you built triangle tables, are the measurements of the table accurate?
- Is the table a right triangle?
- If you completed a dragon tiling, is it accurate?
- Does it show right triangles?
- Offer students feedback as needed.

## Sine and Cosine Ratios

I. Section Objectives

- Review the different parts of right triangles.
- Identify and use the sine ratio in a right triangle.
- Identify and use the cosine ratio in a right triangle.
- Understand sine and cosine ratios in special right triangles.

II. Cross- curricular-Land Surveying

- Find a local land surveyor and ask him/her to visit the classroom.
- This is an opportunity to have a speaker come and teach the students about how geometry can be applied in real life situations.
- Ask the speaker to be prepared to show the connection between land surveying and geometry.
- Also ask him/her to please bring tools for students to see.
- Students will need to complete some research on land surveying prior to the speakers presentation.
- Have the students prepare five questions each to ask the speaker, and be sure that the students ask questions of the speaker when he/she is there.
- Students can prepare a written report sharing how land surveying is connected to geometry following the presentation.

III. Technology Integration

- Use the internet to research land surveying.
- Be sure that the students understand what land surveyors do, some of the tools used, and how right triangles play a part in land surveying.
- There are several websites that students can visit to do this.
- They will find a lot of information simply by using google or Wikipedia.

IV. Notes on Assessment

- Read each students report on the speaker.
- Assess student knowledge and ability to connect this career with geometry.
- Did the students simply repeat the presentation, or did he/she bring their own thoughts and opinions into the paper?
- Provide students with feedback on their work.

## Inverse Trigonometric Ratios

I. Section Objectives

- Identify and use the arctangent ratio in a right triangle.
- Identify and use the arcsine ratio in a right triangle.
- Identify and use the arccosine ratio in a right triangle.
- Understand the general trends of trigonometric ratios.

II. Cross- curricular-Environmental Studies

- Use the following page of information to show students how tangents and arctangents are used in real world applications.
- www.e-education.psu.edu/natureofgeoinfo/c7_p10.html
- This website shows the students that when people are studying the environment and changes in elevation, that they use the measurements of slope to do this.
- Review slope with the students.
- Then you can move on to connecting slope with the tangent ratios.
- These ratios can show how a slope or how elevation is changing over time.
- For example, take beach erosion.
- When the beach or coast is eroding away due to a storm or hurricane, the slope of the land before the storm and after the storm can be compared.
- In the same example, the change in the degrees of the triangle (the arctangent) can be used to compare or demonstrate change as well.
- Have the students think of other types of elevation changes.
- Brainstorm examples and write them on the board.

III. Technology Integration

- Here is a great video showing a word problem and how to figure it out.
- www.video.yahoo.com/watch/3008744/8600194
- Students can watch this video for some extra practice on solving trigonometric word problems.
- Then they can practice writing their own.
- Have the students write an answer key too.
- When finished, collect the word problems for further use.

IV. Notes on Assessment

- Collect student word problems.
- Read them and assess whether or not the students have a good grasp of the material.
- Use the word problems for a quiz or homework assignment.
- Provide students with feedback as needed.

## Acute and Obtuse Triangles

I. Section Objectives

- Identify and use the Law of Sines.
- Identify and use the Law of Cosines.

II. Cross- curricular-Comic Strip

- Students are going to write a comic strip that tells what happens when someone breaks the Law of Sines.
- Students can make this comical and draw characters to go with it.
- It is a creative assignment, but one that also incorporates mathematical information in it.
- It should be considered a fun assignment, but one that also needs to be accurate.
- Students can work on their comic strip in pairs.
- Allow time for students to work.
- Each strip should have writing and animation with it.
- When finished, allow time for the students to share their work.

III. Technology Integration

- Here is a movie that students can watch about landscape architecture and triangulation.
- www.thefutureschannel.com/hands-on_math/survey_team.php
- After watching the film, conduct a discussion on the film and what students learned about geometry and being a landscape architect.

IV. Notes on Assessment

- Collect each comic strip.
- Read them and assess them on two levels.
- 1. Is the mathematical content accurate?
- 2. Is it presented in a creative way?
- Provide students with feedback/correction as needed.

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Feb 22, 2012## Last Modified:

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