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.

Math

Unit 6

11th Grade

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.

Pacing: 16 instructional days (14 lessons, 1 flex day, 1 assessment day)

Assessment

The following assessments accompany Unit 6.

Post-Unit

Use the resources below to assess student understanding of the unit content and action plan for future units.

Post-Unit Assessment

Unit Prep

Intellectual Prep

Suggestions for how to prepare to teach this unit

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

Read "Intuitive Understanding of Sine Waves" on BetterExplained. It explains sine and cosine very well.

Essential Understandings

The central mathematical concepts that students will come to understand in this unit

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

Terms and notation that students learn or use in the unit

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

Materials

The materials, representations, and tools teachers and students will need for this unit

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

Lesson Map

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

Lesson

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

Standards

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

Lesson

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

Standards

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

Lesson

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

Standards

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

Lesson

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

Standards

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

Lesson

Describe the relationship between the unit circle and tangent.

Standards

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

Lesson

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

Standards

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

Lesson

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

Standards

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

Lesson

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

Standards

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

Topic B: Graphing Sine, Cosine, and Target

Lesson

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

Standards

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

Lesson

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

Standards

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

Lesson

Graph transformations of tangent functions.

Standards

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

Lesson

Identify equations and graphs of all six trigonometric functions.

Standards

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

Lesson

Model contextual situations using trigonometric functions (Part I).

Standards

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

Lesson

Model contextual situations using trigonometric functions (Part II).

Standards

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

Common Core Standards

Key

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

The content standards covered in this unit

Building Functions

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

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.
Interpreting Functions

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.
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.

Trigonometric Functions

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.
Trigonometric Functions

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.
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.
Trigonometric Functions

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

Standards covered in previous units or grades that are important background for the current unit

Circles

Circles

G.C.A.2 — Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
Circles

G.C.B.5 — Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Expressing Geometric Properties with Equations

Expressing Geometric Properties with Equations

G.GPE.B.5 — Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
Expressing Geometric Properties with Equations

G.GPE.B.7 — Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. 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.

Geometry

Geometry

7.G.B.4 — Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
Geometry

8.G.B.6 — Explain a proof of the Pythagorean Theorem and its converse.
Geometry

8.G.B.7 — Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
Geometry

8.G.B.8 — Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Similarity, Right Triangles, and Trigonometry

Similarity, Right Triangles, and Trigonometry

G.SRT.C.6 — Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
Similarity, Right Triangles, and Trigonometry

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

G.SRT.C.8 — Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 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.
Similarity, Right Triangles, and Trigonometry

G.SRT.D.10 — Prove the Laws of Sines and Cosines and use them to solve problems.
Similarity, Right Triangles, and Trigonometry

G.SRT.D.11 — Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
Similarity, Right Triangles, and Trigonometry

G.SRT.D.9 — Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

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.

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