Providence Public Schools D-57

Precalculus, Quarter 2, Unit 2.5

Proving Trigonometric Identities
Overview

Number of instruction days: 57 (1 day = 53 minutes)
Content to Be Learned Mathematical Practices to Be Integrated
Verify proofs of Pythagorean identities.
Apply Pythagorean, reciprocal, quotient,
symmetry, and opposite angle identities to
prove other identities.
Prove sum and difference identities for sine,
cosine, and tangent.
2 Reason abstractly and quantitatively.
Make sense of problems and persevere in
proving trigonometric identities.
3 Construct viable arguments and critique the
reasoning of others.
Build a logical progression of statements in
order to confirm trigonometric identities.
6 Attend to precision.
Examine claims and make explicit use of
definitions in verifying trigonometric identities.
Essential Questions
What is the relationship between the
Pythagorean Theorem and the Pythagorean
identities in trigonometry?
Why is the study of trigonometric identities
important?
In what ways can trigonometric identities be
used to simplify trigonometric expressions?

Precalculus, Quarter 2, Unit 2.5 Proving Trigonometric Identities (57 days)
Version 4
D-58 Providence Public Schools

Standards
Common Core State Standards for Mathematical Content
Functions
Trigonometric Functions F-TF
Prove and apply trigonometric identities
F-TF.8 Prove the Pythagorean identity 1 cos sin
2 2
and use it to find sin , cos , or tan
given sin , cos , or tan and the quadrant of the angle.
F-TF.9 (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to
solve problems.
Common Core State Standards for Mathematical Practice
2 Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations.
They bring two complementary abilities to bear on problems involving quantitative relationships: the
ability to decontextualizeto abstract a given situation and represent it symbolically and manipulate the
representing symbols as if they have a life of their own, without necessarily attending to their referents
and the ability to contextualize, to pause as needed during the manipulation process in order to probe into
the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent
representation of the problem at hand; considering the units involved; attending to the meaning of
quantities, not just how to compute them; and knowing and flexibly using different properties of
operations and objects.
3 Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously
established results in constructing arguments. They make conjectures and build a logical progression of
statements to explore the truth of their conjectures. They are able to analyze situations by breaking them
into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them
to others, and respond to the arguments of others. They reason inductively about data, making plausible
arguments that take into account the context from which the data arose. Mathematically proficient
students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or
reasoning from that which is flawed, andif there is a flaw in an argumentexplain what it is.
Elementary students can construct arguments using concrete referents such as objects, drawings,
diagrams, and actions. Such arguments can make sense and be correct, even though they are not
generalized or made formal until later grades. Later, students learn to determine domains to which an
argument applies. Students at all grades can listen or read the arguments of others, decide whether they
make sense, and ask useful questions to clarify or improve the arguments.
6 Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear
definitions in discussion with others and in their own reasoning. They state the meaning of the symbols
Proving Trigonometric Identities (57 days) Precalculus, Quarter 2, Unit 2.5
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Providence Public Schools D-59
they choose, including using the equal sign consistently and appropriately. They are careful about
specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem.
They calculate accurately and efficiently, express numerical answers with a degree of precision
appropriate for the problem context. In the elementary grades, students give carefully formulated
explanations to each other. By the time they reach high school they have learned to examine claims and
make explicit use of definitions.
Clarifying the Standards
Prior Learning
In Algebra I and Geometry, students solved problems algebraically and graphically and solved problems
involving systems of linear equations. Also in Algebra I, students used the Pythagorean Theorem and
literal equations to solve problem situations. In Algebra II, students were introduced to factoring,
completing the square, and solving and interpreting solutions involving polynomial, piecewise, absolute
value, rational, and radical functions. Geometry students used proofs in their studies. Algebra II students
proved the Pythagorean identity 1 cos sin
2 2
and used it in their study of the unit circle.
Current Learning
Precalculus students use trigonometric identities other than the Pythagorean identity for the first time.
Students use the basic trigonometric identities to verify other identities, and they develop and use the sum
and difference identities.
Future Learning
Calculus students will use these identities throughout their work with differentiation and integration
techniques.
Additional Findings
There are no additional findings for this unit.

Assessment
When constructing an end-of-unit assessment, be aware that the assessment should measure your
students understanding of the big ideas indicated within the standards. The CCSS for Mathematical
Content and the CCSS for Mathematical Practice should be considered when designing assessments.
Standards-based mathematics assessment items should vary in difficulty, content, and type. The
assessment should comprise a mix of items, which could include multiple choice items, short and
extended response items, and performance-based tasks. When creating your assessment, you should be
mindful when an item could be differentiated to address the needs of students in your class.
Precalculus, Quarter 2, Unit 2.5 Proving Trigonometric Identities (57 days)
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D-60 Providence Public Schools
The mathematical concepts below are not a prioritized list of assessment items, and your assessment is
not limited to these concepts. However, care should be given to assess the skills the students have
developed within this unit. The assessment should provide you with credible evidence as to your students
attainment of the mathematics within the unit.
Prove the Pythagorean identity sin
2
() + cos
2
() = 1.
Find numerical values of trigonometric functions.
Use the basic identities to verify other trigonometric identities.
Apply trigonometric identities to model real-world context.
Formulate trigonometric identity equations to solve real-world problems.

Instruction
Learning Objectives
Students will be able to:
Prove and use the Pythagorean identity sin
2
() + cos
2
() = 1.
Identify and use reciprocal identities, quotient identities, Pythagorean identities, symmetry identities,
and opposite-angle identities.
Use the basic trigonometric identities to verify other identities.
Use sum and difference identities for the sine, cosine, and tangent functions to solve real world
problems.
Demonstrate understanding of concepts and skills learned in this unit.
Resources
Advanced Mathematical Concepts: Precalculus with Applications, Glencoe, 2006, Teacher Edition
and Student Edition
Section 7-1 (pp. 421-430)
Section 7-2 (pp. 431-436)
Section 7-3 (pp. 437-445) with supplemental material for modeling real-world situations
TeacherWorks All-In-One Planner and Resource Center CD-ROM
Exam View Assessment Suite
Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the
Planning for Effective Instructional Design and Delivery and Assessment sections for specific recommendations.
Materials
TI-Nspire graphing calculators, formula chart (See the Planning for Effective Instructional Design and
Delivery section.
Proving Trigonometric Identities (57 days) Precalculus, Quarter 2, Unit 2.5
Version 4
Providence Public Schools D-61
Instructional Considerations
Key Vocabulary
basic trigonometry identities
quotient identities
opposite-angle identities
Pythagorean identities
symmetry identities
Planning for Effective I nstructional Design and Delivery
Reinforced vocabulary from previous grades or units: cosecant, cosine, cotangent, secant, sine, and
tangent.
Precalculus students are required by the Common Core State Standards to prove the addition and
subtraction formulas for sine, cosine, and tangent. The extension of proofs of trigonometric identities is
considered vital for the depth and rigor of study expected for Precalculus students.
In helping students navigate the complexities of proving trigonometric identities and solving real-world
problems, you can use nonlinguistic representations. A graphic representation in the form of a formula
sheet will provide students with a visual and eventually a mental image to help them remember the basic
formulas to use in proofs or in solving real-world problems. Students can also make their own formula
sheet, leaving room to add completed examples of real-world problems that can be solved using that
particular formula. By adding the completed examples, students will have access to the algebraic
manipulations that must be accomplished to solve trigonometric equations.
Proofs are equality relationships between two mathematical expressions; they are used to simplify
algebraic expressions and to solve algebraic equations. No single method works for all identities.
However, following certain steps might help. To verify an identity, students may start by using basic
identities to transform the more complicated side of the identity into the same expression as the other side.
Students may also use the technique of transforming the two sides of the identity into the same
expression. To do this work, students do not have to have memorized all the identities; their learning
could be supported by the use of a formula sheet with the principal identities.
Have students verify trigonometric identities at the board or using a projection device, explaining their
understanding to other students as they work through the process.
Have students work in pairs or groups of three to verify trigonometric identities on large chart paper. Give
each group a different identity, and be careful to have all different kinds of techniques represented in
the examples that must be verified. Have students share their different ways of accomplishing this task.
After each group has shared their verification, ask the whole group if they have another way to verify the
same identity. Note: The same process could be used in having students solve real-world problems.
Precalculus, Quarter 2, Unit 2.5 Proving Trigonometric Identities (57 days)
Version 4
D-62 Providence Public Schools

Notes