Lesson 8 | Unit Circle and Trigonometric Functions | 11th Grade Mathematics

Objective

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

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

Visualize transformations of sine, cosine, and tangent on the unit circle.
Use transformations of sine, cosine, and tangent to more quickly evaluate particular angles.

Fishtank Plus

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Anchor Problems

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

Problem 1

FInd the value of the expression below:

$${\mathrm{sin}\frac{11\pi}{6}}$$

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

References

EngageNY Mathematics Precalculus and Advanced Topics > Module 4 > Topic A > Lesson 1 — Lesson 1, Exercise 2

Precalculus and Advanced Topics > Module 4 > Topic A > Lesson 1 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

Explain why each of the following expressions are equivalent:

$${\mathrm{sin}\theta=\mathrm{sin}(x-\theta)}$$

$${\mathrm{cos}\theta=\mathrm{cos}(2\pi-\theta)}$$

$${\mathrm{cos}\theta=\mathrm{cos}(-\theta)}$$

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

Target Task

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

Problem 1

Evaluate the following trigonometric expressions and explain how you used the unit circle to determine your answer.

a. $${\mathrm{sin}\left(\pi+\frac{\pi}{3}\right)}$$

b. $${\mathrm{cos}\left(2\pi-\frac{\pi}{6}\right)}$$

References

EngageNY Mathematics Precalculus and Advanced Topics > Module 4 > Topic A > Lesson 1 — Exit Ticket, Question #1

Problem 2

Corinne says that for any real number $${\theta}$$, $$\mathrm{cos}{\theta}=\mathrm{cos}{\theta}-\pi$$. Is she correct? Explain how you know.

References

EngageNY Mathematics Precalculus and Advanced Topics > Module 4 > Topic A > Lesson 1 — Exit Ticket, Question #2

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 evaluating angles close to $${\pi}$$ and $$2{\pi}$$ and ask students to visualize them multiple ways.
Include problems asking about the equivalence of different expressions, as in the second anchor problem.
Include problems evaluating expressions written multiple ways and explaining why they are the same, such as $$\mathrm{cos}\frac{3{\pi}}{4}$$ and $$\mathrm{cos}\frac{5{\pi}}{4}$$.

EngageNY Mathematics Precalculus and Advanced Topics > Module 4 > Topic A > Lesson 1 — Problem Set

Lesson 7

Lesson 9

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

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Target Task

Problem 1

References

Problem 2

References

Additional Practice

Lesson Map

Request a Demo

Contact Information

School Information

What courses are you interested in?

ELA

Math

Are you interested in onboarding professional learning for your teachers and instructional leaders?

Any other information you would like to provide about your school?

Effective Instruction Made Easy