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4.8: 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. Multiple Intelligences

  • This lesson uses the Pythagorean Theorem in several different ways.
  • You can differentiate this lesson by expanding each of the examples in the lesson.
  • Begin by drawing and labeling the parts of right triangle. Be sure that students understand which are the legs and the hypotenuse.
  • Prove the Pythagorean Theorem
  • 1. Start with a right triangle.
  • 2. Construct the altitude
  • 3. Use it in an example.
  • Start with an example where the hypotenuse is missing.
  • Ask students to use the Pythagorean Theorem to find the length of the hypotenuse.
  • Example: leg 1= 4, \mathrm{leg} \ 2 = 6
  • Answer- c = 7.2
  • Be sure that students understand that they will probably need to round to the nearest tenth.
  • Move to finding a missing side length.
  • Example, \mathrm{leg} \ 1= a, \mathrm{leg} \ 2= 4, \mathrm{hypotenuse}= 5
  • Have students solve this for leg a.
  • Answer is 3.
  • Introduce the concept of a Pythagorean Theorem. Show the difference between the first example where we did not have a perfect square and needed to round, and the second example where our answer was a perfect square.
  • Return to the text and demonstrate the other Pythagorean Triples.
  • Move on to finding the area of an isosceles triangle.
  • Walk through this example in the text.
  • Complete the exercise on the board step by step.
  • Then allow time for student questions.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal

III. Special Needs/Modifications

  • Review constructing an altitude.
  • Review symbol for similar.
  • Review finding square roots/radicals.
  • Review the concept of a perfect square.

IV. Alternative Assessment

  • Observe students as they work.
  • Check in periodically throughout the lesson to be sure that students understand the material.
  • Review any information that is not clear.

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. Multiple Intelligences

  • Begin by teaching all of the information in this lesson.
  • You will need to prepare this activity by creating triangles of different sizes to put around the room. Be sure that there are some acute, obtuse and right triangles.
  • Then, let students know that they are going to go on a triangle hunt. They are going to search around the room and locate different triangles.
  • Each student needs to find a triangle and test it out to figure out if the triangle is an acute, obtuse or right triangle.
  • They need to be prepared to justify their answer.
  • The students should repeat this process with three different triangles.
  • When finished, have the students gather in small groups to share their findings.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, bodily- kinesthetic, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • Define the Converse of the Pythagorean Theorem.
  • Review that a converse statement switches the if and the then part of a conditional statement.
  • Write out the formula for finding out whether a triangle is right triangle, an acute triangle or an obtuse triangle.

IV. Alternative Assessment

  • Listen in on the group discussions.
  • Be sure to ask questions and probe into student thinking.
  • Also check each student’s work on the triangles.
  • This will help you to assess the accuracy of the student work.
  • You may want to collect the work for a classwork grade.

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. Multiple Intelligences

  • One way to differentiate this lesson is to have the students teach the concepts in the lesson.
  • You can do this by dividing up the content as follows.
  • 1. Group 1- teaches a review of arithmetic mean.
  • 2. Group 2- teaches geometric mean
  • 3. Group 3- teaches finding the length of an altitude
  • 4. Group 4- teaches finding the length of a leg
  • If you have a large class, you can assign one group the same topic for a different perspective.
  • If you choose to do this activity, DO NOT teach the content first.
  • Assign students the text and let them decipher it.
  • This will also give you an opportunity to observe student understanding.
  • Allow time for group work and request that students use diagrams in their presentations.
  • When finished, each group “teaches” their concept to the others.
  • Allow time for feedback, questions and clarification.
  • Intelligences- logical- mathematical, linguistic, visual- spatial, interpersonal, intrapersonal, bodily- kinesthetic

III. Special Needs/Modifications

  • Inscribed- remind the students of the circles
  • Define altitude.
  • Review definition for similar objects.
  • Review finding the arithmetic mean.
  • Provide students with these notes to help clarify the material.
  • 1. To find the length of the altitude- take the length of the segments of the divided hypotenuse and find the geometric mean.
  • 2. To find the length of the leg- multiple line segment of divided hypotenuse times the length of the hypotenuse and take the square root of the product.

IV. Alternative Assessment

  • Provide feedback during presentations.
  • Assess student learning during group work and presentations.

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. Multiple Intelligences

  • Begin this lesson with an exploration about what happens when you divide up different shapes. Do this before teaching the content of the lesson.
  • Start by having the students draw an equilateral triangle.
  • Pose the question “What happens when you divide an equilateral triangle in half?”
  • Have students actually cut their triangles in half using scissors.
  • Then brainstorm answers to the questions.
  • Then begin a new exploration.
  • Have students draw a square.
  • Pose the question, “What happens when you cut a square in half along the diagonal?”
  • Have students cut their squares along the diagonal using scissors.
  • Then brainstorm answers to the question.
  • You should be able to create two columns of this information on the board.
  • Label one side 45- 45 - 90 and the other side 30- 60- 90
  • Tell students that these are the concepts that you are going to be working with in the lesson.
  • As you teach the lesson, keep referring back to the information that the students have already discovered during the exploration at the beginning of the class.
  • Then move on to the content in the text.
  • Intelligences- linguistic, logical-mathematical, visual- spatial, bodily- kinesthetic, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • List this description of the right isosceles triangle on the board/overhead.
  • Two sides the same length
  • Congruent base angles of 45^\circ.
  • One right angle
  • Review the Pythagorean Theorem.

IV. Alternative Assessment

  • Have students work through the problems in their notebooks as they are covered in the text.
  • Then allow time for questions and answers.

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. Multiple Intelligences

  • To differentiate this lesson, keep it active by including students in designing triangles to determine the tangent ratio.
  • Begin by covering the material in the lesson.
  • When finished, ask students to work with a partner and design three different right triangles.
  • Ask students to measure and label the side lengths of each triangle, and label each angle.
  • Then have the students exchange papers.
  • The students each find the tangent ratios for each angle in each of the three triangles.
  • Once this is completed, ask them to compare their answers with the chart which gives the angle measure for different special triangles.
  • Have the students note if any of the triangles drawn fall into the category of these special triangles.
  • When finished, ask the students to check each other’s work.
  • Allow time for whole class feedback.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal

III. Special Needs/Modifications

  • Write on board “Trigonometric ratios show the relationship between the sides of a triangle and the angles inside it.”
  • Define Tangent Ratio- of an angle is

\mathrm{Tan}x = \frac{\mathrm{Length\ of\ opposite\ side}}{\mathrm{Length\ of\ adjacent\ side}}

  • The x refers to the angle we are focused on.

IV. Alternative Assessment

  • Collect student work and use the triangles to assess student understanding.
  • Listen to student comments following the activity.
  • Allow time for student questions.

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. Multiple Intelligences

  • To differentiate this lesson, teach the material in the lesson and then complete a “working backwards” activity with the students.
  • In the last lesson, we worked with a right triangle and wrote out the tangent ratios for the angles in the triangle.
  • In this lesson, we are going to start with the cosine and sine for the different angles of a right triangle. Then the students need to take these ratios and design a triangle that matches the ratios.
  • In this way, the activity is called “working backwards.”
  • The angles of the triangle are A, B and C.
  • \mathrm{sinA} = \frac{6}{7}
  • \mathrm{sinB} = \frac{5}{7}
  • \mathrm{cosA} = \frac{5}{7}
  • \mathrm{cosB} = \frac{6}{7}
  • Allow time for the students to work with these ratios and draw a right triangle that matches the ratios.
  • When finished, allow time for student sharing.
  • Intelligences- logical- mathematical, linguistic, visual- spatial, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • Review the parts of a triangle.
  • Remind students that the x in the cosine and sine ratios refers to the angle that we are focusing on.
  • Break down the two formulas and write them on the board. Request that the students copy them down in their notebooks.
  • Allow extra time for questions.

IV. Alternative Assessment

  • Collect student work after the activity.
  • Use this to assess student understanding and provide extra support for students who are having difficulty.

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. Multiple Intelligences

  • The best way to differentiate this lesson is to break down the information in the lesson. This will help all students.
  • Here are some notes to give the students as you teach the information in the lesson.
  • Inverse of a trigonometric function has the word arc in front of it.
  • Inverse Tangents
  • Convert measurement to degrees in two ways.
  • 1. Use a table of trigonometric ratios.
  • 2. Use a calculator with “arctan”, “atan” or “tan_1”
  • This will give you the measure of the angle in degrees.
  • Notice that we use the approximately symbol for measurements that are not exact.
  • Point this out for students in the lesson examples.
  • Inverse Sine
  • You can find the arcsine by the same two methods as the arctangent.
  • This converts the measurement to degrees.
  • Inverse Cosine
  • You can find the arccosine the same two ways.
  • This will convert the measurement to degrees.

III. Special Needs/Modifications

  • Begin with some work on inverses.
  • Be sure that students understand an inverse of an operation undoes the operation.
  • Use a one- step equation to show students this.
  • Then use the notes in the Multiple Intelligences section to break down the content for the students.

IV. Alternative Assessment

  • Observe students through this lesson.
  • Allow plenty of time for the students to ask questions.
  • Repeat examples or information that seems unclear.

Acute and Obtuse Triangles

I. Section Objectives

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

II. Multiple Intelligences

  • There is a lot of information in this chapter. I recommend breaking it down and going through the examples slowly so that students are given a visual aid, an auditory aid and a chance to verbally ask questions.
  • Intention of lesson- to apply the sine and cosine ratios to angles in acute and obtuse triangles.
  • Law of Sines- is constant. It can be used to find the missing lengths in triangles.
  • Review using a calculator to find the value of sines.
  • Law of Cosines- works on acute, obtuse and right triangles.
  • If the students seem lost during this lesson, break them up into small groups. Assign each group either the Law of Sines or the Law of Cosines and have them create a poster explaining the steps to using these laws in an example.
  • The students can even use one of the examples in the text.
  • By having the students create a poster to explain the information, the students will learn to assimilate the information themselves.
  • Allow time for each group to explain their poster when finished.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal, intrapersonal

III. Special Needs/Modifications

  • Begin with a review of sines and cosines.
  • Allow time for student questions.
  • Review acute triangles have all angles that are less than 90.
  • Review that obtuse triangles have one angle that is greater than 90.
  • Review using a calculator to find the value of sines.

IV. Alternative Assessment

  • Walk around while the students are working on their posters.
  • Listen to the conversation in the groups.
  • Are the students on task? Are they having difficulty?
  • Often if the conversation has strayed from the content of the assignment, the students are lost and not sure what to do next.
  • Assess student understanding through posters and presentations. Clarify points that have been missed or are incorrect.

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