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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.
  • 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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