Lesson 7 | Unit Circle and Trigonometric Functions | 11th Grade Mathematics

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

Foundational Standards

The foundational standards covered in this lesson

G.C.B.5

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.

Criteria for Success

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

Describe the reasoning behind a “radian.”
Identify the fraction of a circle that is represented by each reference angle and apply that thinking to radians.
Convert radians to degrees and degrees to radians.
Identify sine, cosine, tangent, and the quadrant for any number of radians provided.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Read this blog post "Unit Circles and Radians" by Shireen Dadmehr before beginning the lesson.
Make sure students know how to switch between degrees and radians using their calculators—it will be helpful to mix up problems in the problem set to require students to pay attention to the units.

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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

If $${{{{{360}}}}}$$ degrees is $${2{\pi}}$$ radians, how many degrees are $${\pi}$$ radians?
If $${{{{{360}}}}}$$ degrees is $${2{\pi}}$$ radians, how many degrees are $$\frac{{\pi}}{3}$$ radians?
If $${{{{{360}}}}}$$ degrees is $${2{\pi}}$$ radians, how many radians are $${90}$$ degrees?
If $${{{{{360}}}}}$$ degrees is $${2{\pi}}$$ radians, how many radians are $${540}$$ degrees?
If $${{{{{360}}}}}$$ degrees is $${2{\pi}}$$ radians, how many degrees are $$-\frac{5{\pi}}{4}$$ radians?

Guiding Questions

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Problem 2

How many radii does it take to go around the whole circle? How would you represent this in terms of pi?

Guiding Questions

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References

GeoGebra What is a Radian?

Problem 3

Use the unit circle and indicated triangle below to find the exact value of the sine and cosine of the special angle $${\frac{\pi}{4}}$$.

Guiding Questions

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References

Illustrative Mathematics Special Triangles 1

Special Triangles 1, accessed on Feb. 26, 2018, 1:26 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.

Target Task

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

Problem 1

Convert $${60^\circ}$$ to radians.

References

EngageNY Mathematics Algebra II > Module 2 > Topic A > Lesson 9 — Exit Ticket, Question #1

Algebra II > Module 2 > Topic A > Lesson 9 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..

Problem 2

Convert $${-\frac{\pi}{2}}$$ rad to degrees.

References

EngageNY Mathematics Algebra II > Module 2 > Topic A > Lesson 9 — Exit Ticket, Question #2

Problem 3

Evaluate $${\mathrm{cos}\frac{3\pi}{4}}$$.

Problem 4

If you evaluated $${\mathrm{sin}3.1}$$ on your calculator in radian mode, would the answer be positive or negative? Why?

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 straightforward practice converting between radians and degrees.

Continuous Everywhere But Differentiable Nowhere Trig War
EngageNY Mathematics Algebra II > Module 2 > Topic A > Lesson 9 — Problem Set
Illustrative Mathematics Special Triangles 2
Insert Clever Math Pun Here Trig Addendum: Modified Radian Fraction Cutouts — Fraction circles and a post about how to use fraction circles if students are stuck

Lesson 6

Lesson 8

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

Foundational Standards

Circles

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

References

Problem 3

Guiding Questions

References

Target Task

Problem 1

References

Problem 2

References

Problem 3

Problem 4

Additional Practice

Lesson Map

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