# MA.912.T.1.2

Solve mathematical and real-world problems involving right triangles using trigonometric ratios and the Pythagorean Theorem.

### Clarifications

Clarification 1: Instruction includes procedural fluency with the relationships of side lengths in special right triangles having angle measures of 30°-60°-90° and 45°-45°-90°.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Trigonometry
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Angle
• Equilateral Triangle
• Hypotenuse
• Isosceles Triangle
• Right Triangle

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8, students solved problems involving right triangles using the Pythagorean Theorem. In Geometry, students use their understanding of sine, cosine and tangent to solve mathematical and real-world problems involving right triangles. In later courses, students will extend this knowledge to solve more difficult problems with right triangles, and extend the concept of trigonometric ratios to trigonometric functions on the unit circle and the number line.
• Within the Geometry course, the expectation is to use angle measures given in degrees and not in radians. Additionally, it is not the expectation for students to master the trigonometric ratios of secant, cosecant and cotangent within this course.
• It is customary to use Greek letters to represent angle measures (e.g., Ø, α, β, γ).
• Problem types include cases where some information about the side lengths or angle measures of a right triangle is missing and one must use trigonometric ratios, inverse of trigonometric ratios or Pythagorean Theorem to determine the unknown length(s) or angle measure(s) within a mathematical or real-world context.
• Instruction includes the concept of inverse trigonometric ratios to determine unknown angle measures and how to find these values using technology, including a calculator. Students should have practice using both notations for the inverse trigonometric ratios (sin−1 A or arcsin A; cos−1 A or arccos A; and tan−1 A or arctan A).
• Instruction includes exploring the relationships of the side lengths of special right triangles 45° − 45° − 90° and 30° − 60° − 90°.
• For example, students should realize that the special right triangle 45° − 45° − 90° is an isosceles right triangle. Therefore, two of its angle measures and side lengths are equivalent. So, if a side length is $x$ units, then students can use the Pythagorean Theorem to determine that the hypotenuse is $x$$\sqrt{2}$ units. Additionally, students can make the connection to its trigonometric ratios:  sin 45° = $\frac{\text{1}}{\text{2}}$ (or equivalently $\frac{\text{2}}{\text{2}}$); cos 45° = $\frac{\text{1}}{\text{2}}$ (or equivalently $\frac{\text{2}}{\text{2}}$); and tan 45° = 1.
• For example, students should realize that the special right triangle 30° − 60° − 90° is half of an equilateral triangle. Students can use that knowledge to determine that the shorter leg is one-half the length of the hypotenuse. So, if the shorter leg is $x$ units and the hypotenuse is 2$x$ units, then students can use the Pythagorean Theorem to determine that the other leg is $x$$\sqrt{3}$ units. Additionally, students can make the connection to its trigonometric ratios such as, sin 30° = $\frac{\text{1}}{\text{2}}$; cos 30° = $\frac{\text{1}}{\text{3}}$ and tan 30° = (or equivalently $\frac{\text{3}}{\text{3}}$).

### Common Misconceptions or Errors

• Students may choose the incorrect trigonometric ratio when solving problems.
• Students may misidentify the sides of triangles.
• For example, students may identify the hypotenuse as being the adjacent leg or confuse the adjacent and opposite sides.

• ABCD is a square.

• Part A. What is the measure of segment BD?
• Part B. What is the measure of segment AC?
• Part C. If the measure of segment BD is 14 units, what is the measure of segment BC?

• Part A. A company is requesting equilateral tiles to be made for their new office floor. If the height of the tile is approximately 10.4 inches, what is the length of the sides of the triangle?
• Part B. The same company decides they also want to use half of a square with the side the same length as the height of the equilateral triangle. What is the length of the hypotenuse of the triangle formed from taking half of the square?

### Instructional Items

Instructional Item 1
• The logo of a local construction company contains an equilateral triangle. The height of the triangle is 10 units. What is the length of the measure of each side of the triangle?

Instructional Item 2
• The right triangle ABC is shown. Angle B is the right angle and the length of AB is 1.5 centimeters and the length of BC is 3.1 centimeters.

• Part A. Determine the measure of angles A and C
• Part B. Determine the length of AC.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206310: Geometry (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912070: Access Mathematics for Liberal Arts (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 - 2023, 2023 and beyond (current))
1206315: Geometry for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912065: Access Geometry (Specifically in versions: 2015 - 2022, 2022 and beyond (current))
1207350: Mathematics for College Liberal Arts (Specifically in versions: 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.T.1.AP.2: Given a mathematical and/or real-world problem involving right triangles, solve using trigonometric ratio or the Pythagorean Theorem.

## Related Resources

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

## Formative Assessments

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

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

Pyramid Height:

Students are asked to determine the length of a side of a right triangle in a real-world problem.

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

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

Deriving and Applying the Law of Sines:

Students will be introduced to a derivation of the Law of Sines and apply the Law of Sines to solve triangles.

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

Will You Survive?:

Students are stranded on a desert island and will need to use the law of sines in order to find the quickest path to a rescue vessel.

Note: This is not an introductory lesson for the standard.

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

Wrapping Up Geometry (Surface Area of Triangular Prisms) :

This lesson is designed to take students from recognizing nets of triangular prisms and finding areas of their individual faces, to finding the surface area of triangular prisms.

Type: Lesson Plan

Just Plane Ol' Area!:

Students will construct various figures on coordinate planes and calculate the perimeter and area. Use of the Pythagorean theorem will be required.

Type: Lesson Plan

Estimating Resources:

Using the case study, "Catapult Catastrophe," students will explore the meaning and importance of managing a project’s scope, construction, and cost. Students will be split into groups to brainstorm and create a materials list for the construction of a catapult for a physics project. Groups will then use the materials list to determine a budget for the project. Each group will submit a list of materials required for the project and a budget based on the price of the materials.

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: Experts

What is Fluency?:

What is fluency? What are the ingredients required to become procedurally fluent in mathematics? Dr. Lawrence Gray explores what it means for students to be fluent in mathematics in this Expert Perspectives video.

Type: Perspectives Video: Expert

Oceanography & Math:

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

Type: Perspectives Video: Expert

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. (For this task, United States coins are used, but the task can be adapted for coins from other countries.)

Finding the area of an equilateral triangle:

This problem solving task asks students to find the area of an equilateral triangle. Various solutions are presented that include the Pythagoren theorem and trigonometric functions.

The Lighthouse Problem:

This problem asks students to model phenomena on the surface of the earth by examining the visibility of the lamp in a lighthouse from a boat.

Mt. Whitney to Death Valley:

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

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.

Eratosthenes and the circumference of the earth:

This problem solving task gives an interesting context for implementing ideas from geometry and trigonometry.

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.

As the Wheel Turns:

In this task, students use trigonometric functions to model the movement of a point around a wheel and, through space. Students also interpret features of graphs in terms of the given real-world context.

## MFAS Formative Assessments

Holiday Lights:

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

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).

Perilous Plunge:

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

Pyramid Height:

Students are asked to determine the length of a side of a right triangle in a real-world problem.

River Width:

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

Sine and Cosine:

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

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.

TV Size:

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

Washington Monument:

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

Will It Fit?:

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

## Original Student Tutorials Mathematics - Grades 9-12

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.

## Student Resources

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

## 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

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. (For this task, United States coins are used, but the task can be adapted for coins from other countries.)

Finding the area of an equilateral triangle:

This problem solving task asks students to find the area of an equilateral triangle. Various solutions are presented that include the Pythagoren theorem and trigonometric functions.

The Lighthouse Problem:

This problem asks students to model phenomena on the surface of the earth by examining the visibility of the lamp in a lighthouse from a boat.

Mt. Whitney to Death Valley:

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

Eratosthenes and the circumference of the earth:

This problem solving task gives an interesting context for implementing ideas from geometry and trigonometry.

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.

As the Wheel Turns:

In this task, students use trigonometric functions to model the movement of a point around a wheel and, through space. Students also interpret features of graphs in terms of the given real-world context.

## Parent Resources

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

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. (For this task, United States coins are used, but the task can be adapted for coins from other countries.)

Finding the area of an equilateral triangle:

This problem solving task asks students to find the area of an equilateral triangle. Various solutions are presented that include the Pythagoren theorem and trigonometric functions.

The Lighthouse Problem:

This problem asks students to model phenomena on the surface of the earth by examining the visibility of the lamp in a lighthouse from a boat.

Mt. Whitney to Death Valley:

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

Eratosthenes and the circumference of the earth:

This problem solving task gives an interesting context for implementing ideas from geometry and trigonometry.