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3.8: Right Triangle Trigonometry

Created by: CK-12

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 $33\;\mathrm{feet}$ 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:
• $33 \times 33 = 1089\;\mathrm{feet}$
• $\sqrt{1089} = 33\;\mathrm{feet}$
• 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 $30-60-90$ 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 $90\;\mathrm{feet}$, 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 $90^\circ$ angle.
• Therefore, you can split up the baseball diamond into $45- 45- 90$ triangles.
• $90^2 + 90^2 = c^2$
• $8100 + 8100 = c^2$
• $16200 = c^2$
• $127.2\;\mathrm{feet}$ 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 $45-45-90$ 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.
• 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.

Date Created:

Feb 22, 2012

Feb 23, 2012
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