Right Triangle Geometry - Lesson Plan

• Explore the relationship of the ratios of lengths of sides of right triangles with the Tangent, Sine, and Cosine functions.
• Calculate Tangent, Sine, and Cosine using a calculator.
• Use Sine, Cosine, and Tangent to find missing sides or angles of right triangle.

Teaching Plan (2 Days)

Elements of this emodule can be used for class discussions or individual activities. The content is sorted into three main sections about Tangent, Sine, and Cosine. These sections follow the same pattern for discovering and exploring the relationships between the ratios of sides of right triangles and the trig functions. Because each section is so similar, instructors can leave out some sections or manipilate the order. However, if the module is presented to students in its complete form, we would suggest taking some repeated activities offline to reduce redundancy. Note: The Cosine section does have a few question items about sine and tangent.

INTRODUCTION (15 minutes)

The objective of this section is to introduce students to the possible ratios comparing sides of a right triangle and the vocabulary used.

Using a computer projector, demonstrate how to go to the eNLVM website, go to your school and class, and login. (5 minutes)

Find All Ratios of Sides

(2 Activities) Students discover all the possible side/side ratios of a right triangle using values and variables.
(1 Activity) Students compare the values of all six ratios of similar triangles.

Opposite, Adjacent, Hypotenuse

(2 Activities) The vocabulary for identifying side of right triangles is introduced. Students practice identifying the sides according to the vocabulary presented.

TANGENT (40 Minutes)

The objective of this section is for students to explore the opposite/adjacent ratio, connect it with the tangent function, and use it to find missing lengths or angles in a right triangle.

Explore Tangent

(2 Activities) Students hypothesize, then experiment to see the effects of changing triangle or angle size on the opposite/adjacent ratio.
(2 Activities) Students calculate the opposite/adjacent ratios of a right triangle with a 32º angle. Using these calculations they then find the length of a missing side of a 32º right triangle.
(2 Activities) Students calculate the opposite/adjacent ratios of a right triangle with a 71º angle. Using these calculations they then find the length of a missing side of a 71º right triangle.
(1 Activity) Students calculate the opposite/adjacent ratios of several angles and identify equal angles by finding equal ratios.
(1 Activity) Students create a table for opposite/adjacent ratios of angles measuring multiples of 10.
(1 Activity) Students use a table of opposite/adjacent values of specific angles to find missing lengths of sides of right triangles.

Define and Use Tangent

(1 Activity) The definition of tangent is connected to the opposite/adjacent ratio and formalized as a function.
(2 Activities) Students are given tips for and practice using calculators to find tangent and inverse tangent.
(2 Activities) Students use tangent and inverse tangent to find missing sides or angles of right triangles. Application problems are included.

SINE (40 Minutes)

The objective of this section is for students to explore the opposite/hypotenuse ratio, connect it with the sine function, and use it to find missing lengths or angles in a right triangle.

Explore Sine

(2 Activities) Students hypothesize, then experiment to see the effects of changing triangle or angle size on the opposite/hypotneuse ratio.
(2 Activities) Students calculate the opposite/hypotneuse ratios of a right triangle with a 32º angle. Using these calculations they then find the length of a missing side of a 32º right triangle.
(2 Activities) Students calculate the opposite/hypotneuse ratios of a right triangle with a 71º angle. Using these calculations they then find the length of a missing side of a 71º right triangle.
(1 Activity) Students calculate the opposite/hypotneuse ratios of several angles and identify equal angles by finding equal ratios.
(1 Activity) Students create a table for opposite/hypotneuse ratios of angles measuring multiples of 10.
(1 Activity) Students use a table of opposite/hypotneuse values of specific angles to find missing lengths of sides of right triangles.

Define and Use Sine

(1 Activity) The definition of sine is connected to the opposite/hypotneuse ratio and formalized as a function.
(2 Activities) Students are given tips for and practice using calculators to find sine and inverse sine.
(2 Activities) Students use sine and inverse sine to find missing sides or angles of right triangles. Application problems are included.

COSINE (40 Minutes)

The objective of this section is for students to explore the adjacent/hypotneuse ratio, connect it with the cosine function, and use it to find missing lengths or angles in a right triangle.

Explore Cosine

(2 Activities) Students hypothesize, then experiment to see the effects of changing triangle or angle size on the adjacent/hypotneuse ratio.
(2 Activities) Students calculate the adjacent/hypotneuse ratios of a right triangle with a 32º angle. Using these calculations they then find the length of a missing side of a 32º right triangle.
(2 Activities) Students calculate the adjacent/hypotneuse ratios of a right triangle with a 71º angle. Using these calculations they then find the length of a missing side of a 71º right triangle.
(1 Activity) Students calculate the adjacent/hypotneuse ratios of several angles and identify equal angles by finding equal ratios.
(1 Activity) Students create a table for adjacent/hypotneuse ratios of angles measuring multiples of 10.
(1 Activity) Students use a table of adjacent/hypotneuse values of specific angles to find missing lengths of sides of right triangles.

Define and Use Cosine

(1 Activity) The definition of cosine is connected to the adjacent/hypotneuse ratio and formalized as a function.
(2 Activities) Students are given tips for and practice using calculators to find cosine and inverse cosine.
(2 Activities) Students use cosine and inverse cosine to find missing sides or angles of right triangles. Application problems are included.

QUIZ (15 Minutes)

Students identify the ratio associated with each sign.
Students recognize the functional aspect of the ratios.
Students use calculators to find trig and their inverse values.
Students find the missing sides or angles in right triangles.

Assessment

• Use the quiz for the purpose of summative assessment and grading.

Credits

Correlation to Standards

Correlation to NCTM Standards

Geometry - Standard 2. Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationship.

• Grades 9-12
  • use trigonometric relationships to determine lengths and angle measures.

Correlation to Utah Standards

• Geometry - 2.2.3. Find the missing measures of right triangles.

Correlation to Arizona Standards

• High School, Strand 3, Concept 3, PO 18: Identify the Sine, Cosine, and Tangent ratios of the acute angles of a right triangle.