Unit Circle and Trigonometric Functions

Students discover trigonometry, which allows them to synthesize knowledge of transformations and properties of other functions while offering a new perspective on functions through periodicity.

Unit Summary

High school math sequences typically include trigonometric functions as the final example of functions in the curriculum. Trigonometry offers an opportunity to synthesize knowledge of transformations and properties of other functions while offering a new perspective on functions through periodicity. 

Trigonometry supports calculus, as the trigonometric functions have fascinating relationships through differentiation and integration and offer a great opportunity to practice calculus skills involving rate of change and accumulation.

While trigonometry is necessary for calculus, it also offers avenues to explore for their own sake. Students must connect geometric interpretations of sine and cosine, the unit circle, and the sine and cosine function and look at these three different mathematical structures as examples of the same underlying idea. Trigonometry also offers an opportunity to model periodic contexts in the world and to better understand phenomena around us.

Assessment

This assessment accompanies Unit 6 and should be given on the suggested assessment day or after completing the unit.

Unit Prep

Essential Understandings

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  • Periodic phenomena are modeled by the trigonometric functions of $${f(x)=\mathrm{sin}(x)}$$, $${f(x)=\mathrm{cos(x)}}$$, and $${f(x)=\mathrm{tan(x)}}$$. These functions are characterized by their period, amplitude, midline, and phase shift and can be expressed algebraically in multiple ways.
  • Trigonometric functions represent values on the unit circle, and trigonometric functions and the Pythagorean theorem connect geometric and functional representations of trigonometry.
  • Trigonometric functions are useful in modeling periodic phenomena in the world, and the features of trigonometric functions correspond to useful features of real-world situations.

Vocabulary

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Sine Cosine Tangent Unit circle
Midline Amplitude Period Periodic function
Degrees/Radians Special Right Triangles Secant Cosecant
Cotangent Standard position Initial ray Terminal ray
Sinusoidal curve Odd/Even functions Symmetrical  

Unit Materials, Representations and Tools

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  • Unit circle
  • Interactive graphing software
  • Trigonometry-specific graphing paper is very helpful for this unit

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 “Why This Unit.”
  • Notice the progression of concepts through the 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.

Unit-Specific Intellectual Prep

Lesson Map

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.

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.

Trigonometric Functions
  • F.TF.A.1 — Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

  • F.TF.A.2 — Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

  • F.TF.A.3 — Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π-x, π+x, and 2π-x in terms of their values for x, where x is any real number.

  • F.TF.A.4 — Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

  • F.TF.B.5 — Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. 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.

Foundational Standards

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Circles
  • G.C.A.2

  • G.C.B.5

Expressing Geometric Properties with Equations
  • G.GPE.B.5

  • G.GPE.B.7

Geometry
  • 7.G.B.4

  • 8.G.B.6

  • 8.G.B.7

  • 8.G.B.8

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

  • G.SRT.C.7

  • G.SRT.C.8

  • G.SRT.D.10

  • G.SRT.D.11

  • G.SRT.D.9

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.