Trigonometric Identities and Equations

Students expand on their knowledge of trigonometry, developing a foundation for calculus concepts by expanding their conception of trigonometric functions and looking at connections between functions.

Unit Summary

Unit 7, Trigonometric Identities and Equations, builds on the previous unit on trigonometric functions to expand students’ knowledge of trigonometry. Students develop a foundation for calculus concepts by expanding their conception of trigonometric functions and looking at connections between trigonometric functions. Reasoning flexibly about trigonometric functions and seeing that expressions that look different on the surface can actually act the same on certain domains sets the stage for a study of differentiation and integration, where periodic functions have many useful properties and act as useful tools to study calculus.

Students also apply algebraic techniques to trigonometry. This part of the unit reinforces algebraic skills while also helping students to better understand trigonometric functions graphically and through the unit circle. As students move more flexibly between representations of trigonometric functions, they develop skills in seeing structure in those functions and practice looking at mathematical objects from multiple perspectives and bringing prior knowledge to bear on a new context. This type of relational thinking helps students to see the power of algebraic manipulation and structure in expressions, allowing them to work more flexibly and to see connections more readily.

Assessment

This assessment accompanies Unit 7 and should be given on the suggested assessment day or after completing the unit.

Unit Prep

Intellectual Prep

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Internalization of Standards via the Unit Assessment

  • Take unit assessment. Annotate for: 
    • Standards that each question aligns to
    • Purpose of each question: spiral, foundational, mastery, developing
    • Strategies and representations used in daily lessons
    • Relationship to Essential Questions of unit 
    • Lesson(s) that assessment points to

Internalization of Trajectory of Unit

  • Read and annotate “Unit Summary.”
  • Notice the progression of concepts through unit using the “Unit at a Glance.”
  • Do all target tasks. Annotate the target tasks for: 
    • Essential questions
    • Connection to assessment questions 
  • Answer the essential questions. (In the beginning, submit them to your instructional leader; toward the end, just bring them to the meeting.)

Essential Understandings

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  • Work flexibly with trigonometric identities to rewrite expressions.
  • Define and use inverse trigonometric functions.
  • Solve trigonometric equations, including quadratic equations and equations that require u-substitution.
  • Apply equation-solving knowledge to modeling contexts.
  • Use the Law of Sines and the Law of Cosines to find missing angles and side lengths in acute triangles.

Vocabulary

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Identity Pythagorean identity
Cofunction identities Inverse trig functions (arcsin, arccos, arctan, arcsec, arccsc, arccot)
Reciprocal identities $${\mathrm{sin}^{-1}x}$$ notation for inverse trig functions
Negative angle identities Linear trigonometric equations
$$u$$-substitution Quadratic trigonometric equations
General solution Exact solution
Double angle formula Sum formula
Difference formula Law of Cosines
Law of Sines  

Unit Materials, Representations and Tools

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  • Trig Cheat Sheet by Paul Dawkins (Page 2 has all of the relevant identities for this unit. Students should memorize reciprocal identities and be able to use other identities with a reference when necessary.)

Graphing Calculator Skills

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  • Graphing trig functions/system of trig function and linear function
  • Calculating inverse of trig functions on the calculator
  • Switching between radian and degree mode on the calculator
  • Finding the intersection, value, zero, minimum, maximum
  • Altering the window
  • Using zoom to graph trigonometric functions from $${-{2π}}$$ to $${2π}$$

Lesson Map

Topic A: Basic Trigonometric Identities and Equivalent Expressions

Topic B: Solve Trigonometric Equations

Topic C: Advanced Identities and Solving Trigonometric Equations

Topic D: Applications and Extensions of Trigonometric Functions

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

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Building Functions
  • F.BF.B.3 — Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

  • F.BF.B.4 — Find inverse functions.

  • F.BF.B.4.D — Produce an invertible function from a non-invertible function by restricting the domain.

Creating Equations
  • A.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

  • A.CED.A.4 — Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.

Interpreting Functions
  • F.IF.B.4 — For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

  • F.IF.C.7 — Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

  • F.IF.C.7.E — Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Similarity, Right Triangles, and Trigonometry
  • G.SRT.C.7 — Explain and use the relationship between the sine and cosine of complementary angles.

  • G.SRT.D.10 — Prove the Laws of Sines and Cosines and use them to solve problems.

Trigonometric Functions
  • F.TF.B.6 — Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

  • F.TF.B.7 — Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

  • F.TF.C.8 — Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

  • F.TF.C.9 — Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Foundational Standards

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Arithmetic with Polynomials and Rational Expressions
  • A.APR.D.6

Building Functions
  • F.BF.B.3

Congruence
  • G.CO.A.2

Geometry
  • 8.G.B.6

  • 8.G.B.7

  • 8.G.B.8

Interpreting Functions
  • F.IF.C.8

  • F.IF.C.8.A

Reasoning with Equations and Inequalities
  • A.REI.A.1

Seeing Structure in Expressions
  • A.SSE.A.2

  • A.SSE.B.3

  • A.SSE.B.3.A

Similarity, Right Triangles, and Trigonometry
  • G.SRT.C.6

  • G.SRT.C.8

Trigonometric Functions
  • F.TF.A.2

  • F.TF.A.3

  • F.TF.A.4

  • F.TF.B.5

Standards for Mathematical Practice

  • CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

  • CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

  • CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

  • CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

  • CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

  • CCSS.MATH.PRACTICE.MP6 — Attend to precision.

  • CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

  • CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.