Common Core Math
High School Functions Teaching and Learning Resources
Domain: TF: Trigonometric Functions
Select the cluster for resources on this page:
TFA: Extend the domain of
trigonometric functions using the unit circle
TFB: Model periodic phenomena with
trigonometric functions
TFC: Prove and apply trigonometric
identities
TFA: Extend the domain of
trigonometric functions using the unit circle
Standards:
 TFA.1. Understand radian measure of an angle
as the length of the arc on the unit circle subtended by the angle.
 TFA.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.
 TFA.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 x, π + x, and 2π – x in terms of their
values for x, where x is any real number.
 TFA.4. (+) Use the unit circle to explain
symmetry (odd and even) and periodicity of trigonometric functions.
Technologyenhanced investigations:
Mathwords.com: Math Dictionary:
Key vocabulary for this domain. Use with TFA, TFB, TFC:
Khan Academy:
LearnZillion:
 Lesson set:
Extend
trigonometric functions to all real numbers using the unit
circle. Five video lessons: Understanding the wrapping
function using the unit circle, find trig values for angles
using reference triangles, understand quadrantal angles by
examining x and y values near them, graph f(x) = sinx and g(x) =
cosx using the unit circle, graph trig functions using a
graphing calculator. Aligns with TFA.2.
 Lesson set:
Explain how
the unit circle in the coordinate plane enables the extension of
trigonometric functions to all real numbers. Six video
lessons: Find trig ratios of angles in the coordinate plane,
find values of trig functions using the unit circle, determine
the signs of trig functions by identifying coordinate signs,
evaluate trig functions of angles in all four quadrants; find
sine, cosine, tangent of angles on the x and y axes, determine
the domain of the six trig functions using ratio denominators.
Aligns with TFA.2.
 Lesson set:
Choose
trigonometric functions to model periodic phenomena.
Five video lessons: Graph the sinusoidal functions by plotting
points; stretch and transform by shifting sinusoidal functions
horizontally and vertically; graph tangent, secant, cosecant,
cotangent by drawing sine and cosine graphs; model periodic
phenomena using the trigonometric functions. Aligns with
TFA.4.
Thinking Mathematics:
Teaching Channel Video:
Ferris Wheel: Trigonometric Functions. Lesson objectives:
Model how a trigonometric function describes the relationship of a
Ferris wheel rider as the wheel spins at a constant rate with
relationship to the height of the rider from the ground. Discern
the relationship between the given measure and the period, phase, offset
and amplitude of a cosine function.
Questions for learners to consider are included. Use with BFA.1.b,
BFB.3, and TFA.1.
A. Dendane: Analyze Math:
Walter Fendt: Math Applets:
Sine, Cosine,
and Tangent of an Angle: Use this applet to demonstrate how the
graphs of these functions are generated as a point moves
counterclockwise around the unit circle.
Wolfram Demonstrations Project:
Download the free Wolfram CDF player
to interact with the following manipulatives. Note: Within the
Wolfram Demonstration Project are
4 manipulatives addressing TFA.1,
19 manipulatives for
TFA.2 (all are relevant),
18 manipulatives for
TFA.3, and
16 manipulatives for TFA.4. Among those:
Ron Blond:

The Unit Circle:
Use this applet to examine the unit circle and the
wrapping function. Learners can see values of coordinates on the
unit circle moving counterclockwise around it, comparing radians and
degrees. Exact or approximate values can be shown. This
works well with standard TFA.2.

Elementary Trig Equations:
This applet will help learners to
geometrically connect references angles on a unit circle to
solutions for elementary trig equations with the sine, cosine, and tangent
functions. Students can see how the unit circle connects to
expressing the values of sine, cosine, and tangent for x, π 
x, π + x, and 2π – x,
and thus to solutions of such equations. They can select
radians or degrees. This works well with standard TFA.3.
PatrickJMT: A
Way to Remember the Entire Unit Circle for Trigonometry.
Video.
WiscOnline, Learning Object:
Learn to Count by Common Radian Units: After students learn what a
radian measure is and have derived key values of the coordinates
associates with common radian units (e.g., π/3, π/4, and π/6), this
activity will provide practice finding those values on a unit circle.
Students do not need to know the definitions of the trig functions to do
this activity. This activity works well with standard TFA.3.
Educational Development Center Work
in Maine: Note: See the section for All Resources.
 Trigonometric Functions and the Unit Circle: "Trigonometry.
Functions. Radians. Unit circle. Help students develop a sound
understanding of the three basic trigonometric functions using
the unit circle as a reference, and aids transition of student
thinking to radians."
 Trigonometric Functions and the Unit Circle (2): “Help students
develop a sound understanding of the three basic trigonometric functions
using the unit circle as a reference, and aids transition of student
thinking to radians. This version connects the sine function on the
graph to the unit circle using a line segment. Includes unit circle
reference points every 30 degrees and every 45 degrees.”
 Ferris Wheel Unit Circle: Create a graph of the height of a
seat on a Ferris wheel to explore the sine function and characteristics
of the unit circle. Users can pivot the Ferris wheel to emphasize the
change in height of a seat, show the sine and cosine functions, turn on
coordinates, degrees, and/or radians associated with the eight benchmark
points on the unit circle, and see values for sine, cosine, and tangent
of the seat position.
Multiple Choice:
WiscOnline, Learning Object:
What is a Radian?: This is a short interactive demonstration of the
size of a radian and the number of radians in a circle, accompanied by
three multiple choice problems to test understanding.
Glencoe Online Learning Center, Algebra 2
(2010) selfcheck quiz: Ch. 13, Lesson 2:
Angles and Angle Measure
OpenEd:

Radian measure of an angle: 3 questions that combine
multiple choice and free response. Includes related review
resources. Aligns with TFA.1.

Unit circle extensions: 3 questions that combine multiple
choice and free response. Includes related review
resources. Aligns with TFA.2.

Unit circle/Symmetry of trig functions: 3 questions that combine multiple
choice and free response. Includes related review
resources. Aligns with TFA.4.
Constructedresponse:
Khan Academy: Practice questions with videos.
OpenEd: Special triangles: 4 free response questions. Includes
related review resources. Aligns with TFA.3.
The Math Page: Trigonometry: Aligns with TFA.
 Arc
Length: Lesson introduces the definition of the radian and
includes practice problems. Use with TFA.1.

Radian Measure. Lessons on converting degrees to radians,
radians to degrees, coterminal angles, and the multiples of π with
practice problems. Use with TFA.2.

Analytic Trigonometry: The Unit Circle: Lesson on how right
triangle trig can extend to definitions of the trig functions of
angles drawn on a circle. It includes the "signs" of the trig
functions in each quadrant, quadrantal angles, and the unit circle.
Practice problems are included. Use with TFA.4.

Trig Functions of Any Angle
Performance tasks:
Georgia Standards of Excellence Framework:
 PreCalculus Unit
1: Introduction to Trigonometric Functions. This unit
contains 13 tasks, including a Culminating Task: Graphing Other
Trigonometric Functions. Tasks align to TFA.1, TFA.2,
TFB.5, TFC.8 and IFB.4, IFC.7.
 PreCalculus Unit
2: Trigonometric Functions. This unit
contains two tasks: Right Triangles and the Unit Circle, and Inverse
Trigonometric Functions. Tasks align to BFB.4, BFB.4.d, TFA.3,
TFA.4, TFB.6, TFB.7 and are also related to TFA.2, TFB.5, and
IFC.7.e.
Mathematics Assessment Project: Standards: High School: Functions:
http://map.mathshell.org/stds.php?standardid=1448
Task 427:
Ferris Wheel Aligns with TFA.
Mathematics Vision Project,
Secondary
3 Student Edition:

Module
6: Trigonometric Functions: This module contains 13 classroom tasks.
Function standard TFA.1 is found in tasks 59; TFA.2 is found
in task 3, tasks 59, and task 13. Task 13 also addresses
TFA.3, TFA.4.
Back to top
TFB: Model periodic phenomena with
trigonometric functions
Standards:
 TFB.5. Choose trigonometric functions to model
periodic phenomena with specified amplitude, frequency, and midline.
 TFB.6. (+) Understand that restricting a
trigonometric function to a domain on which it is always increasing or
always decreasing allows its inverse to be constructed.
 TFB.7. (+) Use inverse functions to solve
trigonometric equations that arise in modeling contexts; evaluate the
solutions using technology, and interpret them in terms of the context.
Technologyenhanced investigations:
A. Dendane: Analyze Math:
Sine
Function and
Cosine
Function: HTML5 applets to explore properties of graphs of the sine
function and cosine function. These applets help learners to
understand the meaning of amplitude, frequency, shifts left or right,
and midline by changing parameters in equations of the form f(x) = a
sin (bx + c) + d and of the form f(x) = a cos (bx +
c) + d. The xaxis in each applet is labeled with radians.
Use with TFB.5.
NCTM Illuminations:
Trigonometric Graphing: Learners can investigate
amplitude, period, and phase shift by examining the graphs of various
trigonometric functions (sin, cos, tan, sec, csc, cot) for equations of
the form f(x) = a sin (bx + c) + d. They can change the
function and parameters a, b, c, and d, and choose to display graphs in
either degrees or radians. Use with TFB.5.
Ron Blond (on learnalberta.ca):
Transformations of Sine and Cosine Functions:
This applet helps learners to understand the meaning of amplitude,
frequency, shifts left or right, and midline by changing the parameters a, b, h and k in the trigonometric functions
y = a sin[ b( x  hπ ) ] + k and y = a cos[ b( x  hπ ) ] +
k. Use with TFB.5.
Wolfram Demonstrations Project:
Download the free Wolfram CDF player
to interact with the following manipulatives. Note: Within the
Wolfram Demonstration Project are
7 manipulatives addressing TFB.5 and
3 manipulatives for TFB.6. Among those:
PatrickJMT: Videos on Inverse Trig Functions.
Multiplechoice:
Constructedresponse:
Khan Academy: Practice questions with videos.
NRICH: Tangled Trig Graphs:
Learners are presented multiple sine graphs on the same set of axes and
are challenged to determine the equations from the graphs and how they
might also be written in terms of the cosine function. "The
problem gives the opportunity to investigate reflections, stretches and
translations of curves, and the corresponding effects on equations"
(Teachers' Resources for this problem). Aligns with TFB.5.
Wolfram Demonstrations Project:
Download the free Wolfram CDF player
to interact with the following manipulative:
Inverse Function of a Trigonometric Function Game:
This Demonstration provides a game for the exact
calculation of f^{1}(f(x)) for trigonometric functions. Aligns
with TFB.6 and TFB.7.
Performance tasks:
Georgia Standards of Excellence Framework:
 PreCalculus Unit
1: Introduction to Trigonometric Functions. This unit
contains 13 tasks, including a Culminating Task: Graphing Other
Trigonometric Functions. Tasks align to TFA.1, TFA.2,
TFB.5, TFC.8 and IFB.4, IFC.7.
 PreCalculus Unit
2: Trigonometric Functions. This unit
contains two tasks: Right Triangles and the Unit Circle, and Inverse
Trigonometric Functions. Tasks align to BFB.4, BFB.4.d, TFA.3,
TFA.4, TFB.6, TFB.7 and are also related to TFA.2, TFB.5, and
IFC.7.e.
Illustrative Mathematics: Functions:
Foxes
and Rabbits 2: This task aligns with standard TFB.5.
Mathematics Vision Project,
Secondary
3 Student Edition:
Back to top
TFC: Prove and apply trigonometric
identities
Standards:
 TFC.8. Prove the Pythagorean identity sin^{2}(θ)
+ cos^{2}(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ)
given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
 TFC.9. (+) Prove the addition and subtraction
formulas for sine, cosine, and tangent and use them to solve problems.
Technologyenhanced investigations:
Purple Math:
Trigonometric Identities: This is a list of all the basic
trigonometric identities, which can be used in solving problems.
YouTube:
Pythagorean Identity Proofs: This is a playlist that features
seven videos on how to prove the Pythagorean identity sin^{2}(θ)
+ cos^{2}(θ) = 1 and using it to find sin(θ), cos(θ),
or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the
angle. Aligns with TFC.8.
Khan Academy: Videos posted at YouTube, which align with TFC.9:
McDougal Littell ClassZone, Algebra 2, 2011: Select resources in
Ch. 14 Trigonometric Graphs, Identities, and Equations for interactive practice: Animations (Virtual manipulatives), Games/Vocabulary Flashcards, and PowerPoint lesson
examples, and interactive problems within the eWorkbook. See multiple
choice below for related sections to use within the eWorkbook (Do
section 14.6). Use with TFC.9.
PatrickJMT: Videos using addition and subtraction formulas for
sine, cosine, and tangent to solve problems. Each aligns with TFC.9:
 Identities for Sum and Differences of Sine and
Cosine: Video examples showing use of these formulas: Example 1: sin(75^{o});
Example 2: cos(15^{o});
Example 3: tan(165^{o}) written as sin(165^{o})/
cos(165^{o}).
 Sum and Difference Identities to Simplify an Expression,
Example 2: This
example uses tan(x + y) to simplify tan(x + 4π).
Wolfram Demonstrations Project:
Download the free Wolfram CDF player
to interact with the following manipulatives. Note: Within the
Wolfram Demonstration Project are
3 manipulatives addressing TFC.8 and
9 manipulatives for TFC.9. Among those:

Two visual proofs of a basic trigonometric identity: the
Pythagorean identity. Aligns with TFC.8.

Trig functions for a right triangle: Aligns with TFC.8.
 Cofunction Identities for Sine and Cosine: This is a visual
demonstration using the definition of each function on a right
triangle to show the sine of an angle = cosine of its complement
and the cosine of an angle = sine of its complement. Use
with TFC.9.
 Difference Formula for Cosine: This is a visual
demonstration of the formula cos(AB) = cos(A)cos(B) +
sin(A)sin(B). Take time to justify the steps in the
explanation and label parts sin(A), cos(A), sin(B), cos(B) on
both figures. It's easier to understand the proof from the
figures when angles A and B are not equal. Aligns with
TFC.9.
 Addition Formula for Sine: This is a visual demonstration of
the formula sin(A+B) = sin(A)cos(B) + cos(A)sin(B). Take
time to justify the steps in the explanation and label parts
sin(A), cos(A), sin(B), cos(B) on both figures. It's
easier to understand the proof from the figures when angles A
and B are not equal. Aligns with TFC.9.
Multiplechoice:
Glencoe Online Learning Center, Algebra 2 (2010) selfcheck quizzes: Ch. 14, Lesson 3:
Sum and Difference of Angles Identities
Khan Academy: Practice questions with videos.
McDougal Littell ClassZone, Algebra 2, 2011: Ch. 14.6 Quiz:
Apply Sum and Difference Formulas
OpenEd:

Pythagorean identity: 5 questions that combine multiple
choice and free response. Includes related review resources.
Aligns with TFC.8.

Addition formulae: 5 questions that combine multiple choice
and free response. Includes related review resources. Aligns
with TFC.9.
Constructedresponse:
Khan Academy: Practice questions with videos.
Performance tasks:
Georgia Standards of Excellence Framework:
 PreCalculus Unit
1: Introduction to Trigonometric Functions. This unit
contains 13 tasks, including a Culminating Task: Graphing Other
Trigonometric Functions. Tasks align to TFA.1, TFA.2,
TFB.5, TFC.8 and IFB.4, IFC.7.
 PreCalculus Unit 4: Trigonometric Identities. This unit
contains eight tasks: Proving the Sine Addition and Subtraction Identities,
Proving the Cosine Addition and Subtraction Identities, A Distance Formula
Proof for the Cosine Addition Identity, Proving the Tangent Addition and Subtraction
Identities, DoubleAngles Identities for Sine, Cosine, Tangent; Cosine
DoubleAngle: A Man with Many Identities; Deriving HalfAngle
Identities, and a Culminating Task: How Many Angles Can You Find?
Tasks align to TFC.9.
Back to top Common Core Math:
Intro  HS Functions Domain:
IF  BF  LE  TF 