# Cluster 3: Define trigonometric ratios and solve problems involving right triangles. (Geometry - Major Cluster)Archived

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

General Information
Number: MAFS.912.G-SRT.3
Title: Define trigonometric ratios and solve problems involving right triangles. (Geometry - Major Cluster)
Type: Cluster
Subject: Mathematics - Archived
Domain-Subdomain: Geometry: Similarity, Right Triangles, & Trigonometry

## Related Standards

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MAFS.912.G-SRT.3.AP.6a
Using a corresponding angle of similar right triangles, show that the relationships of the side ratios are the same, which leads to the definition of trigonometric ratios for acute angles.
MAFS.912.G-SRT.3.AP.7a
Explore the sine of an acute angle and the cosine of its complement and determine their relationship.
MAFS.912.G-SRT.3.AP.8a
Apply both trigonometric ratios and Pythagorean Theorem to solve application problems involving right triangles.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

## Formative Assessments

The Sine of 57:

Students are asked to explain what a given sine ratio indicates about a right triangle and if the sine of a specific value varies depending on the right triangle.

Type: Formative Assessment

The Cosine Ratio:

Students are asked to compare the ratio of corresponding sides of two triangles and to explain how this ratio is related to the cosine of a given angle.

Type: Formative Assessment

Sine and Cosine:

Students are asked to explain the relationship between sine and cosine of the acute angles of a right triangle.

Type: Formative Assessment

Right Triangle Relationships:

Students are given the sine and cosine of angle measures and asked to identify the sine and cosine of their complements.

Type: Formative Assessment

Finding Sine:

Students are asked to explain the relationship between sine and cosine of complementary angles.

Type: Formative Assessment

Patterns in the 30-60-90 Table:

Students are asked to use 30-60-90 triangle relationships to observe and explain the relationship between sin 30 and cos 60 (or sin 60 and cos 30).

Type: Formative Assessment

Washington Monument:

Students are asked to find the angle of elevation in a real world situation modeled by a right triangle.

Type: Formative Assessment

Step Up:

Students are asked to explain the relationship among angles in a diagram involving a right triangle and to find one angle of the right triangle.

Type: Formative Assessment

River Width:

Students are asked to find an unknown length in a real world context requiring right triangle trigonometry.

Type: Formative Assessment

Perilous Plunge:

Students are asked to find an unknown length in a real world context requiring right triangle trigonometry.

Type: Formative Assessment

Lighthouse Keeper:

Students are asked to find the difference between two lengths in a real world context requiring right triangle trigonometry.

Type: Formative Assessment

Holiday Lights:

Students are asked to solve a problem in a real world context requiring the use of the Pythagorean Theorem.

Type: Formative Assessment

Will It Fit?:

Students are asked to solve a problem in a real world context using the Pythagorean Theorem.

Type: Formative Assessment

TV Size:

Students are asked to solve a problem in a real world context requiring the use of the Pythagorean Theorem.

Type: Formative Assessment

## Lesson Plans

Similarity and Trigonometry Connections:

The properties of similarity and the corresponding sides of right triangles are used to discover a pattern that leads to the three trigonometric ratios: sine, cosine, and tangent.

Type: Lesson Plan

The Seven Circles Water Fountain:

Students will apply concepts related to circles, angles, area, and circumference to a design situation.

Type: Lesson Plan

Sine and Cosine Relationship between Complementary Angles:

This is a lesson on the relationship between the Sine and Cosine values of Complementary Angles.

Type: Lesson Plan

Sine, Sine, Everywhere a Sine:

Students discover the complementary relationship between sine and cosine in a right triangle.

Type: Lesson Plan

Discovering Trigonometric Ratios:

Students investigate and discover trigonometric ratios by drawing and measuring side lengths for five triangles that have equivalent angle measure. Students collect, analyze, and discuss data to draw conclusions. This is the introductory lesson to facilitate student discovery of trigonometric ratios and allows students to secure a solid foundation before the use of trigonometry to find missing sides. This lesson has students solving application problems by finding an unknown angle based on length measurements.

Type: Lesson Plan

The Copernicus' Travel:

This lesson uses Inverse Trigonometric Ratios to find acute angle measures in right triangles. Students will analyze the given information and determine the best method to use when solving right triangles. The choices reviewed are Trigonometric Ratios, The Pythagorean Theorem, and Special Right Triangles.

Type: Lesson Plan

The Trig Song:

This lesson is a group project activity designed to reinforce the concepts of sine and cosine. The lesson begins with a spiral review of the concepts, which will move into the group project - writing an original song to demonstrate understanding and application of sine and cosine ratios.

Type: Lesson Plan

How Tall am I?:

Students will determine the height of tall objects using three different calculation methods. They will work in groups to gather their data and perform calculations. A whole-class discussion is conducted at the end to compare results and discuss some of the possible errors.

Type: Lesson Plan

This lesson unit is intended to help you assess how well students are able to solve quadratics in one variable. In particular, the lesson will help you identify and help students who have the following difficulties; making sense of a real life situation and deciding on the math to apply to the problem, solving quadratic equations by taking square roots, completing the square, using the quadratic formula, and factoring, and interpreting results in the context of a real life situation.

Type: Lesson Plan

Introduction to Trigonometry:

The video is a brief introduction to or review of trigonometry. The teacher uses a spray bottle to show the difference between opposite, adjacent, and hypotenuse according to the angle in which the bottle is located. Then she touches upon sine, cosine, and tangent. She uses theatrics to introduce the saying "soh-cah-toa" as an acronym for each of the sine, cosine, and tangent.

Type: Lesson Plan

Geometry Problems: Circles and Triangles:

This lesson unit is intended to help you assess how well students are able to use geometric properties to solve problems. In particular, the lesson will help you identify and help students who have the following difficulties solving problems by determining the lengths of the sides in right triangles and finding the measurements of shapes by decomposing complex shapes into simpler ones. The lesson unit will also help students to recognize that there may be different approaches to geometrical problems, and to understand the relative strengths and weaknesses of those approaches.

Type: Lesson Plan

Splash and Learn:

Students will utilize their knowledge about projectiles to devise a method to launch a water balloon so that it lands on a 1 meter square cloth target at least 25 meters away. If they hit the target with the balloon (not just splash a few drops on it), they receive extra credit on the lab.

Type: Lesson Plan

Rockin' Right Triangle Ratios:

Special Right Triangles and the ratios that work when you have to do to learn those ratios for 30-60-90 and 45-45-90 triangles.

Type: Lesson Plan

Are You Pulling My Trig?:

This lesson is an introduction of the application of trigonometric ratios. Students will solve real-world word problems using trigonometric ratios of sine, cosine and tangent.

Type: Lesson Plan

Pythagoras - You Clever Dog:

This lesson starts with an introduction of the Pythagorean Theorem. It introduces vocabulary, formulas and concepts related to right triangles and the use of the Pythagorean Theorem in the real world. Students will learn the basics through real world application.

Type: Lesson Plan

Let's Get "Triggy":

This lesson helps students discover trigonometric ratios and how to apply them to find the measure of sides and angles of a right triangle.  Students will think about problems, discuss concepts with a partner and then share ideas with the class. Students will collaborate and offer supportive coaching to help deepen each other’s understanding.

Type: Lesson Plan

Calculating the Earth-Sun distance using Satellite Observations of a Venus Transit:

Every school child learns that the earth-sun distance is 93 million miles. Yet, determining this distance was a formidable challenge to the best scientists and mathematicians of the 18th and 19th centuries. The purpose of this lesson is to use the 2012 Transit of Venus as an opportunity to work through the mathematics to calculate the earth-sun distance. The only tools needed are basic knowledge of geometry, algebra, and trigonometry. The lesson is self-contained in that it includes all the data needed to work through the exercise.

Type: Lesson Plan

Survey Says... We're Using TRIG!:

This lesson is meant as a review after being taught basic trigonometric functions. It will allow students to see and solve problems from a real-world setting. The Perspectives video presents math being used in the real-world as a multimedia enhancement to this lesson. Students will find this review lesson interesting and fun.

Type: Lesson Plan

## Original Student Tutorial

Around the World with Right Triangles:

Learn how to use trigonometric ratios to find the heights of famous monuments and solve a real-world application in this interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Expert

Oceanography & Math:

A discussion describing ocean currents studied by a physical oceanographer and how math is involved.

Type: Perspectives Video: Expert

## Presentation/Slideshow

The Pythagorean Theorem: Geometry’s Most Elegant Theorem:

This lesson teaches students about the history of the Pythagorean theorem, along with proofs and applications. It is geared toward high school Geometry students that have completed a year of Algebra and addresses the following national standards of the National Council of Teachers of Mathematics and the Mid-continent Research for Education and Learning: 1) Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; 2) Use visualization, spatial reasoning, and geometric modeling to solve problems; 3) Understand and apply basic and advanced properties of the concepts of geometry; and 4) Use the Pythagorean theorem and its converse and properties of special right triangles to solve mathematical and real-world problems. The video portion is about thirty minutes, and with breaks could be completed in 50 minutes. (You may consider completing over two classes, particularly if you want to allow more time for activities or do some of the enrichment material). These activities could be done individually, in pairs, or groups. I think 2 or 3 students is optimal. The materials required for the activities include scissors, tape, string and markers.

Type: Presentation/Slideshow

Coins in a circular pattern:

Using a chart of diameters of different denominations of coins, students are asked to figure out how many coins fit around a central coin.

Finding the area of an equilateral triangle:

This problem solving task asks students to find the area of an equilateral triangle.

Mt. Whitney to Death Valley:

This task engages students in an open-ended modeling task that uses similarity of right triangles.

Shortest Line Segment from a Point P to a Line L:

This is a foundational geometry task designed to provide a route for students to develop some fundamental geometric properties that may seem rather obvious at first glance. In this case, the fundamental property in question is that the shortest path from a point to a line meets the line at a right angle which is crucial for many further developments in the subject.

Seven Circles III:

This provides an opportunity to model a concrete situation with mathematics. Once a representative picture of the situation described in the problem is drawn (the teacher may provide guidance here as necessary), the solution of the task requires an understanding of the definition of the sine function.

Seven Circles I:

This task is intended to help model a concrete situation with geometry. Placing the seven pennies in a circular pattern is a concrete and fun experiment which leads to a genuine mathematical question: does the physical model with pennies give insight into what happens with seven circles in the plane?

Setting Up Sprinklers:

This modeling task involves several different types of geometric knowledge and problem-solving: finding areas of sectors of circles, using trigonometric ratios to solve right triangles, and decomposing a complicated figure involving multiple circular arcs into parts whose areas can be found.

Neglecting the Curvature of the Earth:

This task applies geometric concepts, namely properties of tangents to circles and of right triangles, in a modeling situation. The key geometric point in this task is to recognize that the line of sight from the mountain top towards the horizon is tangent to the earth. We can then use a right triangle where one leg is tangent to a circle and the other leg is the radius of the circle to investigate this situation.

## Teaching Idea

Measuring the Distance to Nearby Stars Using Parallax:

This video provides a very complete and detailed overview of the parallax effect and how it can be used to measure astronomical distances using the tangent function. A number of student activities are presented throughout the 26 minute video, so students can have the opportunity to engage in measuring distances to stars and other local landmarks and can try making the required calculations on their own.The relevance of this concept to other fields, such as surveying, is also noted in the video.

Type: Teaching Idea

## Tutorials

Using Trigonometry to solve for missing information:

This tutorial will show students how to use trigonometry to solve for missing information in right triangles. This video shows worked examples using trigonometric ratios to solve for missing information and evaluate other trigonometric ratios.

Type: Tutorial

Basic Trigonometry:

This tutorial gives an introduction to trigonometry. This resource discusses the three basic trigonometry functions, sine, cosine, and tangent.

Type: Tutorial

Projectile at an angle:

This video discusses how to figure out the horizontal displacement for a projectile launched at an angle.

Type: Tutorial

LSSS Tutorial: Introduction to Vectors and Scalars:

This resource is intended to serve as a concise introduction to vector and scalar quantities for teachers of secondary math and science. It provides definitions of vectors and scalars as well as physical examples of each type of quantity, and also illustrates the differences between these two types of quantities in both one and two dimensions, through determinations of both distance (scalar) and displacement (vector).

Type: Tutorial

## Video/Audio/Animation

MIT BLOSSOMS - The Juice Seller’s Problem:

"This video lesson presents a real world problem that can be solved by using the Pythagorean theorem. The problem faces a juice seller daily. He has equilateral barrels with equal heights and he always tries to empty the juice of two barrels into a third barrel that has a volume equal to the sum of the volumes of the two barrels. This juice seller wants to find a simple way to help him select the right barrel without wasting time, and without any calculations - since he is ignorant of mathematics. The prerequisite for this lesson includes knowledge of the following: the Pythagorean theorem; calculation of a triangle's area knowing the angle between its two sides; cosine rule; calculation of a circle's area; and calculation of the areas and volumes of solids with regular bases. Materials necessary include: equilateral containers of equal heights; sand; and measuring devices. Examples of in-class activities for the breaks between video segments include class discussions, individual calculations and small group problem solving." (from MIT Blossoms' "Pythagoras and the Juice Seller")

Type: Video/Audio/Animation

## Virtual Manipulative

Demonstrate the Pythagorean Theorem:

Representation to illustrate the Pythagorean Theorem.

Type: Virtual Manipulative

## Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

## Original Student Tutorial

Around the World with Right Triangles:

Learn how to use trigonometric ratios to find the heights of famous monuments and solve a real-world application in this interactive tutorial.

Type: Original Student Tutorial

## Presentation/Slideshow

The Pythagorean Theorem: Geometry’s Most Elegant Theorem:

This lesson teaches students about the history of the Pythagorean theorem, along with proofs and applications. It is geared toward high school Geometry students that have completed a year of Algebra and addresses the following national standards of the National Council of Teachers of Mathematics and the Mid-continent Research for Education and Learning: 1) Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; 2) Use visualization, spatial reasoning, and geometric modeling to solve problems; 3) Understand and apply basic and advanced properties of the concepts of geometry; and 4) Use the Pythagorean theorem and its converse and properties of special right triangles to solve mathematical and real-world problems. The video portion is about thirty minutes, and with breaks could be completed in 50 minutes. (You may consider completing over two classes, particularly if you want to allow more time for activities or do some of the enrichment material). These activities could be done individually, in pairs, or groups. I think 2 or 3 students is optimal. The materials required for the activities include scissors, tape, string and markers.

Type: Presentation/Slideshow

Coins in a circular pattern:

Using a chart of diameters of different denominations of coins, students are asked to figure out how many coins fit around a central coin.

Finding the area of an equilateral triangle:

This problem solving task asks students to find the area of an equilateral triangle.

Mt. Whitney to Death Valley:

This task engages students in an open-ended modeling task that uses similarity of right triangles.

Shortest Line Segment from a Point P to a Line L:

This is a foundational geometry task designed to provide a route for students to develop some fundamental geometric properties that may seem rather obvious at first glance. In this case, the fundamental property in question is that the shortest path from a point to a line meets the line at a right angle which is crucial for many further developments in the subject.

Seven Circles III:

This provides an opportunity to model a concrete situation with mathematics. Once a representative picture of the situation described in the problem is drawn (the teacher may provide guidance here as necessary), the solution of the task requires an understanding of the definition of the sine function.

Seven Circles I:

This task is intended to help model a concrete situation with geometry. Placing the seven pennies in a circular pattern is a concrete and fun experiment which leads to a genuine mathematical question: does the physical model with pennies give insight into what happens with seven circles in the plane?

Setting Up Sprinklers:

This modeling task involves several different types of geometric knowledge and problem-solving: finding areas of sectors of circles, using trigonometric ratios to solve right triangles, and decomposing a complicated figure involving multiple circular arcs into parts whose areas can be found.

Neglecting the Curvature of the Earth:

This task applies geometric concepts, namely properties of tangents to circles and of right triangles, in a modeling situation. The key geometric point in this task is to recognize that the line of sight from the mountain top towards the horizon is tangent to the earth. We can then use a right triangle where one leg is tangent to a circle and the other leg is the radius of the circle to investigate this situation.

## Tutorials

Using Trigonometry to solve for missing information:

This tutorial will show students how to use trigonometry to solve for missing information in right triangles. This video shows worked examples using trigonometric ratios to solve for missing information and evaluate other trigonometric ratios.

Type: Tutorial

Basic Trigonometry:

This tutorial gives an introduction to trigonometry. This resource discusses the three basic trigonometry functions, sine, cosine, and tangent.

Type: Tutorial

Projectile at an angle:

This video discusses how to figure out the horizontal displacement for a projectile launched at an angle.

Type: Tutorial

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.

Coins in a circular pattern:

Using a chart of diameters of different denominations of coins, students are asked to figure out how many coins fit around a central coin.

Finding the area of an equilateral triangle:

This problem solving task asks students to find the area of an equilateral triangle.

Mt. Whitney to Death Valley:

This task engages students in an open-ended modeling task that uses similarity of right triangles.

Shortest Line Segment from a Point P to a Line L:

This is a foundational geometry task designed to provide a route for students to develop some fundamental geometric properties that may seem rather obvious at first glance. In this case, the fundamental property in question is that the shortest path from a point to a line meets the line at a right angle which is crucial for many further developments in the subject.

Seven Circles III:

This provides an opportunity to model a concrete situation with mathematics. Once a representative picture of the situation described in the problem is drawn (the teacher may provide guidance here as necessary), the solution of the task requires an understanding of the definition of the sine function.

Seven Circles I:

This task is intended to help model a concrete situation with geometry. Placing the seven pennies in a circular pattern is a concrete and fun experiment which leads to a genuine mathematical question: does the physical model with pennies give insight into what happens with seven circles in the plane?