Lesson 14 | Unit Circle and Trigonometric Functions | 11th Grade Mathematics

Objective

Model contextual situations using trigonometric functions (Part II).

Common Core Standards

Core Standards

The core standards covered in this lesson

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.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.

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.

Criteria for Success

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

Describe the meaning of parameters for sine and cosine functions in context.
Verify and describe the model in the context of the situation.
Define alternative models that may be used for a contextual situation.

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Anchor Problems

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

Have students participate in the Desmos activity Burning Daylight.

Guiding Questions

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References

Desmos Burning Daylight

Target Task

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

Tidal data for New Canal Station, located on the shore of Lake Pontchartrain, LA and Lake Charles, LA are shown below.

New Canal Station on Lake Pontchartrain, LA, Tide Chart

Date	Day	Time	Height	High/Low
2014/05/28	Wed.	07:22 a.m.	0.12	L
2014/05/28	Wed.	07:11 p.m.	0.53	H
2014/05/29	Thurs.	07:51 a.m.	0.11	L
2014/05/29	Thurs.	07:58 p.m.	0.53	H

Lake Charles, LA, Tide Chart

Date	Day	Time	Height	High/Low
2014/05/28	Wed.	02:20 a.m.	-0.05	L
2014/05/28	Wed.	10:00 a.m.	1.30	H
2014/05/28	Wed.	03:36 p.m.	0.98	L
2014/05/28	Wed.	07:05 p.m.	1.11	H
2014/05/29	Thurs.	02:53 a.m.	-0.06	L
2014/05/29	Thurs.	10:44 a.m.	1.31	H
2014/05/29	Thurs.	04:23 p.m.	1.00	L
2014/05/29	Thurs.	07:37 p.m.	1.10	H

a. Would a sinusoidal fucntion of the form $${f(x)=A\mathrm{sin}(\omega (x-h))+k}$$ be appropriate to model the given data for each location? Explain your reasoning.

b. Write a sinusoidal function to model the data for New Canal Station.

References

EngageNY Mathematics Algebra II > Module 2 > Topic B > Lesson 13 — Exit Ticket

Algebra II > Module 2 > Topic B > Lesson 13 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..

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.

EngageNY Mathematics Algebra II > Module 2 > Topic B > Lesson 13 — Problem Set
Illustrative Mathematics As The Wheel Turns

Lesson 13

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

Trigonometric Functions

Trigonometric Functions

Criteria for Success

Anchor Problems

Guiding Questions

References

Target Task

References

Additional Practice

Lesson Map

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