# Common Core Math

## Domain: TF: Trigonometric Functions

• TF-A: Extend the domain of trigonometric functions using the unit circle

• TF-B: Model periodic phenomena with trigonometric functions

• TF-C: Prove and apply trigonometric identities

### TF-A: Extend the domain of trigonometric functions using the unit circle

Standards:

• TF-A.1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
• 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.
• 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 x, π + x, and 2π – x in terms of their values for x, where x is any real number.
• TF-A.4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Technology-enhanced investigations:

Mathwords.com: Math Dictionary: Key vocabulary for this domain.  Use with TF-A, TF-B, TF-C:

 amplitude inverse function Pythagorean identities unit circle trig definitions cosine inverse trig function radian trig functions domain odd function and even function sine trig identities frequency of periodic function periodic function tangent trig values of special angles

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 TF-A.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 TF-A.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 TF-A.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 BF-A.1.b, BF-B.3, and TF-A.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 TF-A.1, 19 manipulatives for TF-A.2 (all are relevant), 18 manipulatives for TF-A.3, and 16 manipulatives for TF-A.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 TF-A.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 TF-A.3.

PatrickJMT: A Way to Remember the Entire Unit Circle for Trigonometry.  Video.

Wisc-Online, 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 TF-A.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:

Wisc-Online, 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) self-check 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 TF-A.1.
• Unit circle extensions: 3 questions that combine multiple choice and free response.  Includes related review resources.  Aligns with TF-A.2.
• Unit circle/Symmetry of trig functions: 3 questions that combine multiple choice and free response.  Includes related review resources.  Aligns with TF-A.4.

Constructed-response:

Khan Academy: Practice questions with videos.

OpenEd: Special triangles: 4 free response questions.  Includes related review resources.  Aligns with TF-A.3.

The Math Page: Trigonometry: Aligns with TF-A.

• Arc Length: Lesson introduces the definition of the radian and includes practice problems.  Use with TF-A.1.
• Radian Measure.  Lessons on converting degrees to radians, radians to degrees, coterminal angles, and the multiples of π with practice problems.  Use with TF-A.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 TF-A.4.
• Trig Functions of Any Angle

Georgia Standards of Excellence Framework:

Mathematics Assessment Project: Standards: High School: Functions: http://map.mathshell.org/stds.php?standardid=1448 Task 427: Ferris Wheel  Aligns with TF-A.

Mathematics Vision Project, Secondary 3 Student Edition:

### TF-B: Model periodic phenomena with trigonometric functions

Standards:

• TF-B.5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
• TF-B.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.
• TF-B.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.

Technology-enhanced 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 x-axis in each applet is labeled with radians. Use with TF-B.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 TF-B.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 TF-B.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 TF-B.5 and 3 manipulatives for TF-B.6.  Among those:

PatrickJMT: Videos on Inverse Trig Functions.

Multiple-choice:

OpenEd:

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulative: Elementary Transformations of a Sine Wave Quiz: Learners are shown a sine graph and a transformation of it. They select parameters to create the equation of the transformation.  Aligns with TF-B.5.

Constructed-response:

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 TF-B.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 TF-B.6 and TF-B.7.

Georgia Standards of Excellence Framework:

Illustrative Mathematics: Functions: Foxes and Rabbits 2: This task aligns with standard TF-B.5.

Mathematics Vision Project, Secondary 3 Student Edition:

### TF-C: Prove and apply trigonometric identities

Standards:

• TF-C.8. Prove the Pythagorean identity sin2(θ) + cos2(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.
• TF-C.9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Technology-enhanced 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 sin2(θ) + cos2(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 TF-C.8.

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 TF-C.9.

PatrickJMT: Videos using addition and subtraction formulas for sine, cosine, and tangent to solve problems.  Each aligns with TF-C.9:

• Identities for Sum and Differences of Sine and Cosine: Video examples showing use of these formulas:  Example 1: sin(75o); Example 2: cos(15o); Example 3: tan(165o) written as sin(165o)/ cos(165o).
• 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 TF-C.8 and 9 manipulatives for TF-C.9.  Among those:

• Two visual proofs of a basic trigonometric identity: the Pythagorean identity.  Aligns with TF-C.8.
• Trig functions for a right triangle: Aligns with TF-C.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 TF-C.9.
• Difference Formula for Cosine: This is a visual demonstration of the formula cos(A-B) = 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 TF-C.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 TF-C.9.

Multiple-choice:

Glencoe Online Learning Center, Algebra 2 (2010) self-check 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 TF-C.8.
• Addition formulae: 5 questions that combine multiple choice and free response.  Includes related review resources.  Aligns with TF-C.9.

Constructed-response:

Khan Academy: Practice questions with videos.