Curriculum / Math / 11th Grade / Unit 6: Unit Circle and Trigonometric Functions / Lesson 6

Unit Circle and Trigonometric Functions

Lesson 6

Math

Unit 6

11th Grade

Lesson 6 of 14

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

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.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.
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.SRT.C.6

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.

Criteria for Success

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

Define cosecant, secant, and cotangent in terms of sine, cosine, and tangent.
Evaluate all six trigonometric functions at reference angles.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

This Demos tool with the unit circle is helpful for demonstration purposes.
"10 Secret Trig Functions your Math Teachers Never Taught You" by Evelyn Lamb on ScientificAmerican.com provides interesting context for the different trigonometric functions.

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

25-30 minutes

Problem 1

Below are six different trigonometric ratios, all noted on the unit circle.

“$${\mathrm{sec}\theta}$$” is “secant of theta”
“$${\mathrm{cosec}\theta}$$” is “cosecant of theta” (also can be written as $${\mathrm{csc}\theta}$$)
“$${\mathrm{cot}\theta}$$” is “cotangent of theta”

What is the relationship between the sine, cosine, and tangent and these three new trigonometric functions?

Guiding Questions

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

References

GeoGebra Reciprocal Trig Functions on the Unit Circle

Reciprocal Trig Functions on the Unit Circle by mathsurgery.wikispaces is made available by GeoGebra under the CC BY-NC-SA 3.0 license. Copyright © International GeoGebra Institute, 2013. Accessed Feb. 26, 2018, 1 p.m..

Problem 2

Evaluate the following expressions:

a. $${\mathrm{csc}(30^\circ)}$$

b. $${\mathrm{cot}(135^\circ)}$$

c. $${\frac{\mathrm{cos}(45^\circ)}{\mathrm{sec}(60^\circ)}}$$

d. $${-\mathrm{cos}(-60^\circ)}$$

Guiding Questions

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

Problem 3

The image below shows where the values of sine and cosine are positive and negative. Fill out a similar image for cosecant, secant, tangent, and cotangent.

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

5-10 minutes

Problem 1

Evaluate:

$${\mathrm{tan}0^\circ}$$

$${\mathrm{cot}45^\circ}$$

$${\mathrm{csc}90^\circ}$$

Problem 2

Where is the value of $${\mathrm{sec}\theta}$$ undefined?

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.

Open Middle Trig Ratios
Open Middle Trig Functions
EngageNY Mathematics Algebra II > Module 2 > Topic A > Lesson 7
Illustrative Mathematics Equilateral Triangles and Trigonometric Functions — This is an additional problem that uses some geometric concepts to get at trigonometry.
Math Mistakes If sin(x)/cos(x)=3/5, then 1/cos(x)=1/5 — This post is awesome. It gets at the ratio reasoning and using properties too literally without remembering that they are ratios.

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

Lesson 7

Contact Information

First Name *

Last Name *

Email *

Phone *

Job Title *

School Information

School or District Name *

Address *

Address 2

City *

State *

Zip *

Country *

Average # of Students Per Grade *

Start Date *

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

Yes

Unit Circle and Trigonometric Functions

Objective

Add a note

Are You Sure?

Common Core Standards

Core Standards

Trigonometric Functions

Trigonometric Functions

Foundational Standards

Similarity, Right Triangles, and Trigonometry

Criteria for Success

Add a note

Are You Sure?

Tips for Teachers

Add a note

Are You Sure?

Anchor Problems

Add a note

Are You Sure?

Problem 1

Add a note

Are You Sure?

Guiding Questions

References

Problem 2

Add a note

Are You Sure?

Guiding Questions

Problem 3

Add a note

Are You Sure?

Guiding Questions

Target Task

Add a note

Are You Sure?

Problem 1

Add a note

Are You Sure?

Problem 2

Add a note

Are You Sure?

Additional Practice

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?

We Handle Materials So You Can Focus on Students