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 Instructional Focus DocumentPrecalculus
 TITLE : Unit 07: Problem Solving with Trigonometric Ratios SUGGESTED DURATION : 10 days

Unit Overview

This unit bundles student expectations that address using trigonometric relationships such as special right triangles, right triangle trigonometry, the Law of Sines, and the Law of Cosines, to solve for missing sides and angles. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this unit, in Geometry Unit 04, students applied the Triangle Inequality Theorem to solve problems and proved two triangles were congruent using congruence conditions. In Geometry Unit 05, students determined side lengths of right triangles using the Pythagorean Theorem and the relationships of special right triangles (30°– 60°– 90° and 45°– 45°– 90°). Additionally, students used the trigonometric ratios (sine, cosine, and tangent) to determine triangle side lengths and angle measures.

During this unit, students determine the values of trigonometric ratios for right triangles (including sine, cosine, tangent, cosecant, secant, and cotangent) and use these ratios to solve mathematical and real-world problems involving finding sides and angles. Problem situations involve both special angles and non-special angles. Students solve problems involving trigonometric ratios on the coordinate plane, including determining the values of trigonometric ratios and solving mathematical and real-world problems. Students use trigonometry in real-world problems to explore the connections between standard position rotation angles and directional (navigational) bearings. Students recognize when the Law of Sines and the Law of Cosines can be applied and use the Law of Sines in mathematical and real-world problems, including problems that utilize directional bearings. Students use trigonometry (including right triangle trigonometry, the Law of Sines, and the Law of Cosines) to solve mathematical and real-world problems and recognize which trigonometric equation is applicable for a given problem situation.

After this unit, in Precalculus Units 08 – 12, students will continue to apply trigonometric relationships when studying trigonometric functions, trigonometric equations and identities, vectors, parametric equations, and polar equations. In subsequent mathematics courses, students will continue to apply these concepts as they arise in problem situations.

Geometric reasoning serves an integral role in college readiness. Focusing on geometric figures, their properties, and their connections to other mathematical content areas is emphasized in the Texas College and Career Readiness Standards (TxCCRS): III. Geometric Reasoning A3; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

According to the National Council of Teachers of Mathematics (2000), instruction in grades 9 – 12 should enable all students to “analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships” including using trigonometric ratios to determine lengths and angle measures (p. 308). Geometric ideas, including the use of trigonometric ratios, are useful in other areas of mathematics, as well as in applied settings (NCTM, 2000). In Texas, the importance of these skills is emphasized in the Texas College and Career Readiness Standards (2009), which calls for students to be able to recognize and apply right triangle relationships including basic trigonometry. A fundamental experience in high school trigonometry includes familiarity with similar triangles and the “length ratio properties that are invariant across them” (Sinclair, Pimm, & Skelin, 2012, p. 70). In the AP Calculus Course Description, the College Board (2012) states that mathematics designed for college-bound students should involve the study of advanced topics in trigonometry.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
Sinclair, N., Pimm, D., & Skelin, M. (2012). Developing essential understanding of geometry for teaching mathematics in grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.

OVERARCHING UNDERSTANDINGS and QUESTIONS

Application of attributes and measures of figures can be generalized to describe geometric relationships which can be used to solve problem situations.

• Why are attributes and measures of figures used to generalize geometric relationships?
• How can numeric patterns be used to formulate geometric relationships?
• Why is it important to distinguish measureable attributes?
• How do geometric relationships relate to other geometric relationships?
• Why is it essential to develop generalizations for geometric relationships?
• How are geometric relationships applied to solve problem situations?
• How do different systems of measure relate to one another?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Multiple Representations
• Patterns/Rules
• Ratios
• Solve

Geometric Reasoning

• Geometric Attributes/Properties
• Geometric Relationships
• Trigonometric Relationships

Measurement Reasoning

• Angle Measures
• Formulas
• Length

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Geometric tools such as special right triangle relationships, trigonometric ratios, the Law of Sines, and the Law of Cosines are used to determine lengths and angles in triangles.

• How can the special right triangle relationships be used to calculate distances?
• For which particular geometric figures can special right triangle relationships be used?
• What information is needed to use special right triangle relationships to solve problems?
• How can trigonometric ratios be used to calculate distances and angle measures in a right triangle?
• For what particular figures can trigonometric ratios be used to solve problems?
• What information is needed to use trigonometric ratios to solve problem situations?
• How can the Law of Sines be used to calculate distances and angle measures in a triangle?
• For what particular figures can the Law of Sines be used to solve problems?
• What information is needed to use the Law of Sines to solve problem situations?
• Under what conditions does the Law of Sines result in an ambiguous case?
• How can the Law of Cosines be used to calculate distances and angle measures in a triangle?
• For what particular figures can the Law of Cosines be used to solve problems?
• What information is needed to use the Law of Cosines to solve problem situations?

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that inverse trigonometric functions are the same as the reciprocals of the trigonometric functions. For example, sin–1(x) is not the same as .
• Some students may incorrectly evaluate trigonometric functions or inverse trigonometric functions if they use a calculator in the wrong angle mode.
• Some students may think that angles of elevation and depression are angles formed from the slant line and the vertical, instead of the angles formed by the slant line with the horizontal.

Underdeveloped Concepts:

• Some students may think that a triangle with acute angle θ where cos(θ) =  must have an adjacent side of length 3 and a hypotenuse of length 5. However, the trigonometric ratio cos(θ) =  gives the ratio of the adjacent side and hypotenuse and not necessarily their specific measures.
• Some students may not consider the presence of a second possible solution (e.g., the ambiguous case) when given the measures of two sides in a triangle and a non-included acute angle.
• Some students may think that the Law of Sines and the Law of Cosines apply only to non-right triangles. Instead, these trigonometric relationships can be used in both non-right and right triangles.
• Students may think that a bearing starts with the positive x-axis as zero instead of starting at positive y-axis and may think the rotation is counterclockwise instead of clockwise.

Unit Vocabulary

• Special right triangles – right triangles which have angles that measure 30°– 60°– 90 or 45°– 45°– 90° degrees
• Trigonometric functions – a function of an angle expressed as the ratio of two of the sides of a right triangle that contain that angle
• Trigonometric ratios – a ratio of the measures of two sides of a right triangle based on their position in relation to an acute angle in the right triangle

Related Vocabulary:

 Adjacent Ambiguous case Angle Angle of depression Angle of elevation Bearing Cosecant Cosine Cotangent Distance Hypotenuse Inverse trigonometric functions Law of sines Law of cosines Leg Length Opposite Pythagorean Theorem Pythagorean triple Reciprocal Special angles Special right triangles Tangent Trigonometric function Trigonometric ratio Trigonometry
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Creator if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards (select CCRS from Standard Set dropdown menu)

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Precalculus Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)
• Blue text: Supporting information / Clarifications from TCMPC (Specificity)
• Blue text in italics: Unit-specific clarification
• Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)
P.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
P.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE
Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
P.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
P.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS
Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
P.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE
Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
P.1E Create and use representations to organize, record, and communicate mathematical ideas.

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
P.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
P.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
P.2 Functions. The student uses process standards in mathematics to explore, describe, and analyze the attributes of functions. The student makes connections between multiple representations of functions and algebraically constructs new functions. The student analyzes and uses functions to model real-world problems. The student is expected to:
P.2P Determine the values of the trigonometric functions at the special angles and relate them in mathematical and real-world problems.

Determine

THE VALUES OF THE TRIGONOMETRIC FUNCTIONS AT THE SPECIAL ANGLES

Including, but not limited to:

• Trigonometric functions – a function of an angle expressed as the ratio of two of the sides of a right triangle that contain that angle
• Trigonometric functions are called circular functions because they are based on the unit circle (circle with radius of 1 unit and center at the origin) and right triangle relationships within the circle.
• Sine (sin x or sin )
• Cosine (cos x or cos )
• Tangent (tan x or tan )
• Cotangent (cot x or cot )
• Secant (sec x or sec )
• Cosecant (csc x or csc )
• Special angles
• Degrees
• Multiples of 30°
• Multiples of 60°
• Multiples of 45°
• Special right triangles – right triangles which have angles that measure 30°– 60°– 90° or 45°– 45°– 90°
• In a 45°– 45°– 90° triangle, the proportions of the sides are 1x:1x:x.
• In a 30°– 60°– 90° triangle, the proportions of the sides are 1x:x:2x.
• Special angles as reference angles

Relate

THE VALUES OF TRIGONOMETRIC FUNCTIONS AT THE SPECIAL ANGLES IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Mathematical problem situations
• Real-world problem situations

Note(s):

• Geometry determined side lengths of right triangles using the Pythagorean Theorem and the relationships of special right triangles (30°– 60°– 90° and 45°– 45°– 90°).
• Geometry used the trigonometric ratios (sine, cosine, and tangent) to determine triangle side lengths and angle measures.
• Precalculus extends trigonometric ratios from angles in triangles to include any real radian measure.
• Precalculus generalizes the relationships of special right triangles to determine the values of the trigonometric functions of any multiple of 30° or 45°.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• A3 – Recognize and apply right triangle relationships including basic trigonometry.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.4 Number and measure. The student uses process standards in mathematics to apply appropriate techniques, tools, and formulas to calculate measures in mathematical and real-world problems. The student is expected to:
P.4E Determine the value of trigonometric ratios of angles and solve problems involving trigonometric ratios in mathematical and real-world problems.

Determine

THE VALUE OF TRIGONOMETRIC RATIOS OF ANGLES

Including, but not limited to:

• Right triangles
• If △ABC has a right angle at C
• Trigonometric ratios – a ratio of the measures of two sides of a right triangle based on their position in relation to an acute angle in the right triangle

• Right triangles on the coordinate grid for any radius value
• Trigonometric functions of an angle in standard position whose terminal side passes through the point (xy)
• sin  = , where r =
• cos  = , where r =
• tan  =
• cot  =
• sec  = , where r =
• csc  = , where r =

Solve

PROBLEMS INVOLVING TRIGONOMETRIC RATIOS

Including, but not limited to:

• Mathematical problem situations involving right triangles
• Finding side lengths and angle measures of right triangles
• Determining trigonometric values
• Applying trigonometric values
• Applying the Pythagorean Theorem
• Determining measures of right triangles on a coordinate grid for any radius value
• Real-world problem situations
• Problems involving right triangles
• Problems involving distances and angles
• Problems involving angles of elevation and depression

Note(s):

• Geometry used the trigonometric ratios (sine, cosine, and tangent) to determine triangle side lengths and angle measures.
• Precalculus extends trigonometric ratios from angles in triangles to include any real radian measure or degree measure.
• Precalculus generalizes the relationships in right triangles to determine the values of the trigonometric functions for any real angle measure.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• A3 – Recognize and apply right triangle relationships including basic trigonometry.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.4F Use trigonometry in mathematical and real-world problems, including directional bearing.

Use

TRIGONOMETRY IN MATHEMATICAL AND REAL-WORLD PROBLEMS, INCLUDING DIRECTIONAL BEARING

Including, but not limited to:

• Mathematical problem situations involving right triangles
• Finding side lengths and angle measures of right triangles
• Determining and applying trigonometric values
• Determining measures of right triangles on a coordinate grid for any radius value
• Determining measures of rotation angles
• Real-world problem situations
• Problems involving right triangles
• Problems involving distances and angles
• Problems involving angles of elevation and depression
• Problems involving directional bearing
• Angle measure in degrees between 0° and 360°, determined by a clockwise rotation from the north line
• Angle measure in degrees between 0° and 90°, determined by a rotation east or west of the north-south line
• Relationship between standard position angles, true (navigational) bearings, and conventional (quadrant) bearings
• Tools to use when solving trigonometric problems
• Pythagorean Theorem
• Pythagorean triples
• Trigonometric ratios
• Right triangle relationships
• Special right triangle relationships
• Law of Sines and/or the Law of Cosines
• Formula for Law of Sines
• Formulas for Law of Cosines
• a2 = b2c2 – 2bc cos A
• b2 = a2 + c2 – 2ac cos B
• c2 = a2 + b2 – 2ab cos C

Note(s):

• Geometry used the trigonometric ratios (sine, cosine, and tangent) to determine side lengths and angle measures in right triangles.
• Precalculus extends the use of the trigonometric ratios to solve real-world problems, including those involving navigational bearings.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• A3 – Recognize and apply right triangle relationships including basic trigonometry.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.4G Use the Law of Sines in mathematical and real-world problems.

Use

THE LAW OF SINES IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Law of Sines
• Recognizing when the Law of Sines can be applied to solve for side lengths and angle measures in a triangle
• Given two angles and the included side of a triangle (or “ASA”), determine one of the remaining side lengths.
• Given two angles and a non-included side of a triangle (or “AAS”), determine one of the remaining side lengths.
• Given two sides and a non-included angle of a triangle (or “SSA”), determine one of the remaining angle measures.
• Analyzing both solutions to triangles where the Law of Sines produces an ambiguous case
• Solving various types of mathematical and real-world problem situations
• Mathematical problems
• Real-world problems

Note(s):

• Geometry used the trigonometric ratios (sine, cosine, and tangent) to determine side lengths and angle measures in right triangles.
• Precalculus extends the use of the trigonometric ratios to solve for side lengths and angle measures in certain triangles using the Law of Sines.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• A3 – Recognize and apply right triangle relationships including basic trigonometry.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.4H Use the Law of Cosines in mathematical and real-world problems.

Use

THE LAW OF COSINES IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Law of Cosines
• a2 = b2 + c2 – 2bc cos A
• b2 = a2 + c2 – 2ac cos B
• c2 = a2 + b2 – 2ab cos C
• Recognizing when the Law of Cosines can be applied to solve for side lengths and angle measures in a triangle
• Given three sides of a triangle (or “SSS”), determine one of the angle measures.
• Given two sides and the included angle of a triangle (or “SAS”), determine the length of the remaining side.
• Solving various types of mathematical and real-world problem situations

Note(s):