Lesson 9 | Unit Circle and Trigonometric Functions | 11th Grade Mathematics

Objective

Graph transformations of sine and cosine functions (Part I).

Common Core Standards

Core Standards

The core standards covered in this lesson

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

Interpreting Functions

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

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

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.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.

Trigonometric Functions

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.

Trigonometric Functions

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

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Describe how the graphs of sine and cosine functions relate to the unit circle.
Identify amplitude, period, and midline of sine and cosine functions.
Identify maximum and minimum values of sine and cosine functions.
Explain why the sine function is odd and the cosine function is even.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

This GeoGebra tool, Unit Circle and Sine Graph, draws the graph from the unit circle.
Use radians for any materials from here on out even though the referenced Algebra 2 EngageNY lesson has the material in degrees.
Focus on getting students practice with simple transformations of sine and cosine functions.

Fishtank Plus

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

The points on the graphs and the unit circle below were chosen so that there is a relationship between them.

Explain the relationship between the coordinates $$a$$, $$b$$, $$c$$, and $$d$$ marked on the graph of $$y=\sin(t) $$ and $$y=\cos(t)$$ and the quantities $$A$$, $$B$$, and $$C$$ marked in the diagram of the unit circle below.

Guiding Questions

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References

Illustrative Mathematics Trig Functions and the Unit Circle

Trig Functions and the Unit Circle, accessed on Feb. 26, 2018, 3:25 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem 2

Of sine and cosine, one function is even and the other is odd. Why is this the case?

Guiding Questions

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Problem 3

Match each graph with its equation.

$${y=\mathrm{sin}x}$$

$${y=2\mathrm{sin}x}$$

$${y=\mathrm{sin}(2x)}$$

$${y=\mathrm{sin}x}+2$$

Guiding Questions

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References

New Visions for Public Schools Sine Graphs and Equations

Sine Graphs and Equations is made available by New Visions for Public Schools under the CC BY-NC-SA 4.0 license. © 2017 New Visions for Public Schools. Accessed https://curriculum.newvisions.org/math/resources/resource/algebra-ii-unit-3-big-idea-4-sine-graphs-and-equations/.

Modified by Fishtank Learning, Inc.

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Problem 1

Sketch a graph of the sine function on the interval $${0<x<2\pi}$$ showing all key points of the graph (vertical and horizontal intercepts and maximum and minimum points). Mark the coordinates of the maximum and minimum points and the intercepts.

References

EngageNY Mathematics Algebra II > Module 2 > Topic A > Lesson 8 — Exit Ticket, Question #1

Algebra II > Module 2 > Topic A > Lesson 8 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

Sketch a graph of the cosine function on the interval $${0<x<2\pi}$$ showing all key points of the graph (vertical and horizontal intercepts and maximum and minimum points). Mark the coordinates of the maximum and minimum points and the intercepts.

References

EngageNY Mathematics Algebra II > Module 2 > Topic A > Lesson 8 — Exit Ticket, Question #2

Modified by Fishtank Learning, Inc.

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Mark the coordinate points on the graph for the values in the unit circle.
Include problems graphing simple transformations of sine and cosine function, as in the third anchor problem.

EngageNY Mathematics Algebra II > Module 2 > Topic A > Lesson 8 — Problem Set
EngageNY Mathematics Precalculus and Advanced Topics > Module 4 > Topic A > Lesson 1 — Discussion, "Symmetry of Sine, Cosine, and Tangent Functions" (Skip the tangent function)

Lesson 8

Lesson 10

Lesson Map

Topic A: Trigonometric Ratios in Application and on the Unit Circle

Sketch sine graphs to model contextual situations and identify features of sine graphs.

F.IF.B.4 F.IF.C.7.E F.TF.A.4 F.TF.B.5

Sketch cosine graphs to model contextual situations and identify features of cosine graphs.

F.IF.B.4 F.IF.C.7.E F.TF.B.5

Find values of specific points on the unit circle using geometry.

F.TF.A.2 F.TF.A.4

Evaluate sines and cosines of points at reference angles on the unit circle.

F.TF.A.2 F.TF.A.4

Describe the relationship between the unit circle and tangent.

F.TF.A.2 F.TF.A.3

Define and evaluate the trigonometric functions of tangent, cosecant, secant, and cotangent.

F.TF.A.2 F.TF.A.3

Convert between degrees and radians and evaluate trigonometric functions written in radians.

F.TF.A.1 F.TF.A.3

Evaluate transformations of sine, cosine, and tangent such as $${2{\pi-x}}$$, $${\pi-x}$$, and $${\pi+x}$$.

F.TF.A.1 F.TF.A.3

Topic B: Graphing Sine, Cosine, and Target

Graph transformations of sine and cosine functions (Part I).

F.IF.C.7.E F.TF.A.1 F.TF.A.3 F.TF.A.4

Graph transformations of sine and cosine functions (Part II).

F.BF.B.3 F.IF.C.7.E

Graph transformations of tangent functions.

F.BF.B.3 F.IF.C.7.E

Identify equations and graphs of all six trigonometric functions.

F.BF.B.3 F.TF.A.3 F.TF.B.5

Model contextual situations using trigonometric functions (Part I).

F.TF.A.3 F.TF.B.5

Model contextual situations using trigonometric functions (Part II).

F.TF.A.3 F.TF.B.5

Unit Circle and Trigonometric Functions

Objective

Common Core Standards

Core Standards

Interpreting Functions

Trigonometric Functions

Trigonometric Functions

Trigonometric Functions

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Problem 3

Guiding Questions

References

Target Task

Problem 1

References

Problem 2

References

Additional Practice

Lesson Map

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