Lesson 12 | Unit Circle and Trigonometric Functions | 11th Grade Mathematics

Objective

Identify equations and graphs of all six trigonometric functions.

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

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.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

Use features of trigonometric functions to identify corresponding graphs and descriptions.
Establish reasoning for why an equation matches with one graph but not another.
Recognize secant, cosecant, and cotangent graphs as reciprocals of the other trig functions.
Critique reasoning of others and describe flawed reasoning when matching features, functions, and graphs.

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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

Show students the word doc linked in the blog post Sinusoidal Sort by Chris Hunter.

Ask, "Which functions, descriptions, and graphs match together?"

References

Reflections in the Why Sinusoidal Sort

Sinusoidal Sort by Chris Hunter made available on Reflection in the Why under a CC BY-NC 4.0 license. Accessed Feb. 26, 2018, 4:10 p.m..

Problem 2

In each graph below, a function is graphed as a solid line and its reciprocal is graphed as a dotted line. What features of each function can you connect to its reciprocal?

Based on what you notice, sketch graphs of $${y=\mathrm{sec}x}$$ and $${y=\mathrm{csc}x}$$.

Guiding Questions

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Target Task

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

Which one of the equations below matches the graph?

A) $${y=-3\mathrm{sin}(4x)}$$

B) $${y=3\mathrm{cos}(4x)}$$

C) $${y=3\mathrm{sin}\left(\frac{1}{4}x\right)}$$

D) $${y=3\mathrm{cos}\left(\frac{1}{4}x\right)}$$

References

Kelly Pernell Trigonometry - Math 50 - Graphs of Trigonometric Functions - Homework 2 — Question #1

Trigonometry - Math 50 - Graphs of Trigonometric Functions - Homework 2 made available by Kelly Pernell at http://www.berkeleycitycollege.edu/wp/kpernell/files/2013/02/13spr_m50_hwprobs_2.pdf. Accessed Aug. 3, 2018, 11:45 a.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.

Include problems recognizing secant and cosecant graphs. Students don’t need to sketch graphs themselves, but they should recognize them, connect them with sine and cosine graphs, and understand the reciprocal relationship.
Include problems connecting transformations of cosecant and secant functions to transformations of sine and cosine functions.
Includes review of graphing all trigonometric functions.
Include problems writing equations and sketching graphs from a list of characteristics.

Math Mistakes Amplitude and Period — This is a good error analysis problem
Desmos Match My Trig Function [Radians]

Lesson 11

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

Building Functions

Trigonometric Functions

Trigonometric Functions

Criteria for Success

Anchor Problems

Problem 1

References

Problem 2

Guiding Questions

Target Task

References

Additional Practice

Lesson Map

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