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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
• 2. Construct the altitude
• 3. Use it in an example.
• 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.
• 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 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.

## Date Created:

Feb 22, 2012

Feb 23, 2012
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