# 4.8: 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. 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
- Answer
- Be sure that students understand that they will probably need to round to the nearest tenth.
- Move to finding a missing side length.
- Example,
- Have students solve this for leg a.
- Answer is .
- 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 and the other side
- 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 .
- 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

- The 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 and .
- 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 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 .
- Review that obtuse triangles have one angle that is greater than .
- 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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Feb 22, 2012## Last Modified:

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