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Standard 1 : Define and use trigonometric ratios, identities or functions to solve problems.
Cluster Standards

This cluster includes the following benchmarks.

Visit the specific benchmark webpage to find related instructional resources.

  • MA.912.T.1.1 : Define trigonometric ratios for acute angles in right triangles.
  • MA.912.T.1.2 : Solve mathematical and real-world problems involving right triangles using trigonometric ratios and the Pythagorean Theorem.
  • MA.912.T.1.3 : Apply the Law of Sines and the Law of Cosines to solve mathematical and real-world problems involving triangles.
  • MA.912.T.1.4 : Solve mathematical problems involving finding the area of a triangle given two sides and the included angle.
  • MA.912.T.1.5 : Prove Pythagorean Identities. Apply Pythagorean Identities to calculate trigonometric ratios and to solve problems.
  • MA.912.T.1.6 : Prove the Double-Angle, Half-Angle, Angle Sum and Difference formulas for sine, cosine, and tangent. Apply these formulas to solve problems.
  • MA.912.T.1.7 : Simplify expressions using trigonometric identities.
  • MA.912.T.1.8 : Solve mathematical and real-world problems involving one-variable trigonometric ratios.
Cluster Information
Number:
MA.912.T.1
Title:
Define and use trigonometric ratios, identities or functions to solve problems.
Type:
Standard
Subject:
Mathematics (B.E.S.T.)
Grade:
912
Strand
Trigonometry
Cluster Access Points

This cluster includes the following Access Points.

  • MA.912.T.1.AP.1 : Select a trigonometric ratio for acute angles in right triangles limited to sine or cosine.
  • 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.
Cluster Resources

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

Original Student Tutorial
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.

  • 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.

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

  • Right Triangle Relationships: Students are given the sine and cosine of angle measures and asked to identify the sine and cosine of their complements.

  • Finding Sine: Students are asked to explain the relationship between sine and cosine of complementary angles.

  • 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).

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

  • Washington Monument: Students are asked to find the angle of elevation in a real world situation modeled by 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.

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

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

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

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

  • Will It Fit?: Students are asked to solve a problem in a real world context using the Pythagorean Theorem.

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

Lesson Plans
  • 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.

  • 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.

  • The Laws of Sine and Cosine: In this lesson, students determine when to use the Law of Cosine and/or the Law of Sine to calculate the measures of sides and angles of non-right triangles.

  • The Seven Circles Water Fountain: Students will apply concepts related to circles, angles, area, and circumference to a design situation.

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

  • Sine, Sine, Everywhere a Sine: Students discover the complementary relationship between sine and cosine in a right triangle.

  • 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.

  • 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.

  • 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.

  • 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.

  • Following the Law of Sine: This lesson introduces the law of sine. It is designed to give students practice in using the law to guide understanding. The summative assessment requires students to use the law of sine to plan a city project.

  • 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.

  • 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. 

  • 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.

  • 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.

  • 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.

  • 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.

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.

  • Oceanography & Math: <p>A discussion describing ocean&nbsp;currents studied by a physical oceanographer and how math is involved.&nbsp;</p>
Perspectives Video: Professional/Enthusiast
Presentation/Slideshow
  • Pythagoras' Theorem: This resource can be used to introduce the Pythagorean Theorem to students. It provides sketches, applets, examples and easy-to-understand visual proofs as well as an algebra proof for the theorem.

    It also includes interactive multiple choice practice questions on solving for a side of a right triangle, and applications involving right triangles, as well as a hands-on activity for students to do that allows them to create their own proof.

Problem-Solving Tasks
  • 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.

  • 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.

  • 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.