Trigonometric Functions

Tim Brzezinski, نصر الدين ابو الهيجاء, May 11, 2020

Table of Contents

  1. Working with Radians
    1. Copy of 1 Radian: Clear Definition
    2. Animation 69
    3. Radian Illustrator
    4. Movie: Radians to Revs
    5. DEG to RAD and RAD to DEG Illustrator
    6. DEG to RAD and RAD to DEG: Question Generator
  2. Angles in Standard Position, Coterminal Angles, Trig Ratios
    1. Angles in Standard Position
    2. Animation 176
    3. Angle Orientations (Standard Position)
    4. Animation 180
    5. Animation 181
    6. Coterminal Angles Action!!!
    7. Animation 183
    8. Quiz: Sketching Angles in Standard Position
    9. Coterminal Angles: Question Generator
    10. Defining Trigonometric Ratios
    11. Unit Circle and Sine Graph
    12. Unit Circle and Tangent Graph
    13. Unit Circle and Cosine Graph
  3. Trig Functions of Special Angles
    1. Right Triangle Generator for Right Triangle Trigonometry
    2. Quiz: Evaluating Trigonometric Functions of Angles Given a Point on its Terminal Ray
    3. Evaluating Trig Functions Given a Point on the Terminal Ray
    4. Quiz: Quadrant Logic (Trig)
    5. Trig Quadrant Logic (2)
    6. Quiz: Evaluating Trig FNS of Angles (Quadrant Logic): V1
    7. Quiz: Evaluating Trig FNS of Angles WITH Quadrant Logic (V2)
    8. Right Triangle Trigonometry: Intro
    9. True Meaning of Sine, Cosine, Tangent Ratios within Right Triangles
    10. Right Triangle Solver: Formative Assessment
    11. Right Triangle Trig: Solving for Sides (2)
    12. Quiz: Expressing Angles Using Inverse Trigonometric Function
    13. Trig Ratio Estimations (Right Triangle Context)
    14. Trig Ratio Estimations (Quiz: V1)
    15. Trig Ratio Estimations (Quiz: V2)
    16. Isosceles Right Triangle: Quick Investigation
    17. Special Right Triangle (I)
    18. Another Special Right Triangle (Guided Discovery)
    19. Special Right Triangle (II)
    20. 30-60-90 Triangle
    21. Special Right Triangles: Basic Questions
    22. Quiz: Special Right Triangles
    23. Trig Function Values (30 Degrees)
    24. Animation 148
    25. Trig Function Values (45 Degrees)
    26. Animation 146
    27. Trig Function Values (60 Degrees)
    28. Animation 147
    29. Special Angles in Standard Position: Intro (V1)
    30. Special Angles in Standard Position: Intro (V2)
    31. Special Angles in Standard Position: Intro (V2) - Part 2
    32. Animation 177
    33. Animation 178
    34. Special Angles in Standard Position: Intro (V3)
    35. Animation 182
    36. Quiz 1: Trigonometric Functions of Special Angles
    37. Quiz 2: Trigonometric Functions of Special Angles
  4. Even and Odd Functions
    1. Sine and Cosecant Functions (Special Property)
    2. Sine Identity
    3. Animation 150
    4. Cosine and Secant Functions (Special Property)
    5. Cosine Identity
    6. Animation 149
    7. Tangent and Cotangent Functions (Special Property)
    8. Animation 145
    9. Tangent Identity
    10. Animation 151
  5. Graphing and Writing Trig Functions
    1. Unit Circle to Sine and Cosine Functions
    2. Sine & Cosine Period Action (1)!
    3. Animation 190
    4. Animation 191
    5. Sine & Cosine Period Action (2)!
    6. Tangent Function Exploration
    7. Tangent Period Action (1)!
    8. Tangent Period Action (2)!
    9. Animation 192
    10. Cotangent Function Exploration
    11. Transforming Sine and Cosine Functions (1): Dynamic Illustrator
    12. Transforming Sine and Cosine Functions (2): Dynamic Illustrator
    13. Transforming the Sine Function (All Transformations)
    14. Graphing Sine & Cosine Functions (II)
    15. Graphing Sine & Cosine Functions (I)
    16. Transformations of a Sinusoid
    17. Quiz: Graphing Sine & Cosine Functions (Amplitude & Vertical Shift Only)
    18. Writing Sine and Cosine Functions (with Amplitude & Vertical Shift Changes)
    19. Sine Function Modeling (y = a sin(x) + d) Quiz
    20. Cosine Function Modeling (y = a cos(x) + d) Quiz
    21. Modeling with Trigonometric Functions
    22. Sine & Cosine Function Modeling: Finding Correct Parameters
    23. Ferris Wheel Action!!!
  6. Inverse Trig Functions
    1. Inverse Relations: Graphs
    2. Sine Function Domain Restriction Options?
    3. Cosine Function Domain Restriction Options?
    4. Tangent Function Domain Restriction Options?
    5. Inverse Trig Functions: Intuitive Explorations
  7. Solving Trig Equations and Modeling
    1. Ferris Wheel (1): Modeling with Trigonometric Functions
    2. Ferris Wheel (2): Modeling with Trigonometric Functions
  8. Proving Trig Identities
    1. New Trig IDs From Similar Right Triangles
    2. New Trig ID's from Similar Right Triangles (V2)
    3. Pythagorean Trigonometric Identity (1)
    4. Trig ID Movie (II)
    5. Trig ID Movie (III)
  9. Proving Advanced Trig Identities
    1. Sine & Cosine of a Sum: Discovery
    2. Trig ID: Geometric Investigation
    3. Tangent of a Sum: Discovery
    4. Tangent of a Difference: Trig Identity Puzzle + Proof!
    5. Cotangent of a Sum: Discovery
    6. Bizzare Trig Identity?
    7. Animation 105
    8. sin(75) & cos(75) activity
  10. Other Functions Resources
    1. Functions Resources

1.

Working with Radians

  • 1. Copy of 1 Radian: Clear Definition
  • 2. Animation 69
  • 3. Radian Illustrator
  • 4. Movie: Radians to Revs
  • 5. DEG to RAD and RAD to DEG Illustrator
  • 6. DEG to RAD and RAD to DEG: Question Generator

1.1.

Copy of 1 Radian: Clear Definition

[color=#c51414]One unit of ANGLE or ARC MEASURE which you're probably familiar with is that of a "degree." One degree is 1/360th of a full revolution, right? [/color][br][color=#0a971e]Another unit of ANGLE or ARC MEASURE is a "revolution". 1 revolution = 360 degrees, right? [/color][br][br][color=#1551b5]Well, there is ANOTHER unit of ANGLE or ARC MEASURE with which you'll soon become familiar. [/color] [br][color=#1551b5]This new unit of ANGLE or ARC MEASURE is called a [b]RADIAN[/b]. [/color] [br][br][i][color=#b20ea8]Interact with the applet below for a few minutes. [br]Reset it a few times and start the animation again each time.[br]Be sure to change the circle's radius as you go along. [br][br][b][color=#1551b5]After interacting with this applet, answer the question that appears immediately below it.[/color][/b][/color] [/i]
Again, recall that a "degree", a "revolution", and a "radian" are all units of ARC MEASURE (i.e. AMOUNT OF SPIN). [br][br][color=#c51414][b]Complete the following sentence definition:[/b][/color] [br][br][b][color=#1551b5]Definition: 1 RADIAN is defined to be a unit of ARC MEASURE for which.....[/color][/b]
– نصر الدين ابو الهيجاء, Tim Brzezinski

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2.

Angles in Standard Position, Coterminal Angles, Trig Ratios

  • 1. Angles in Standard Position
  • 2. Animation 176
  • 3. Angle Orientations (Standard Position)
  • 4. Animation 180
  • 5. Animation 181
  • 6. Coterminal Angles Action!!!
  • 7. Animation 183
  • 8. Quiz: Sketching Angles in Standard Position
  • 9. Coterminal Angles: Question Generator
  • 10. Defining Trigonometric Ratios
  • 11. Unit Circle and Sine Graph
  • 12. Unit Circle and Tangent Graph
  • 13. Unit Circle and Cosine Graph

2.1.

Angles in Standard Position

The angle drawn below in the coordinate plane is classified as being drawn in [b]STANDARD POSITION. [br][br][/b]Interact with the applet for a minute.[br]Then answer the question that follows.
ANGLE IN STANDARD POSITION:
1.
What does it mean for an angle drawn in the coordinate plane to be drawn in [b]STANDARD POSITION? [br][br][/b](Your definition should list 2 criteria.)
– Tim Brzezinski

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3.

Trig Functions of Special Angles

  • 1. Right Triangle Generator for Right Triangle Trigonometry
  • 2. Quiz: Evaluating Trigonometric Functions of Angles Given a Point on its Terminal Ray
  • 3. Evaluating Trig Functions Given a Point on the Terminal Ray
  • 4. Quiz: Quadrant Logic (Trig)
  • 5. Trig Quadrant Logic (2)
  • 6. Quiz: Evaluating Trig FNS of Angles (Quadrant Logic): V1
  • 7. Quiz: Evaluating Trig FNS of Angles WITH Quadrant Logic (V2)
  • 8. Right Triangle Trigonometry: Intro
  • 9. True Meaning of Sine, Cosine, Tangent Ratios within Right Triangles
  • 10. Right Triangle Solver: Formative Assessment
  • 11. Right Triangle Trig: Solving for Sides (2)
  • 12. Quiz: Expressing Angles Using Inverse Trigonometric Function
  • 13. Trig Ratio Estimations (Right Triangle Context)
  • 14. Trig Ratio Estimations (Quiz: V1)
  • 15. Trig Ratio Estimations (Quiz: V2)
  • 16. Isosceles Right Triangle: Quick Investigation
  • 17. Special Right Triangle (I)
  • 18. Another Special Right Triangle (Guided Discovery)
  • 19. Special Right Triangle (II)
  • 20. 30-60-90 Triangle
  • 21. Special Right Triangles: Basic Questions
  • 22. Quiz: Special Right Triangles
  • 23. Trig Function Values (30 Degrees)
  • 24. Animation 148
  • 25. Trig Function Values (45 Degrees)
  • 26. Animation 146
  • 27. Trig Function Values (60 Degrees)
  • 28. Animation 147
  • 29. Special Angles in Standard Position: Intro (V1)
  • 30. Special Angles in Standard Position: Intro (V2)
  • 31. Special Angles in Standard Position: Intro (V2) - Part 2
  • 32. Animation 177
  • 33. Animation 178
  • 34. Special Angles in Standard Position: Intro (V3)
  • 35. Animation 182
  • 36. Quiz 1: Trigonometric Functions of Special Angles
  • 37. Quiz 2: Trigonometric Functions of Special Angles

3.1.

Right Triangle Generator for Right Triangle Trigonometry

Math Teachers and Students:
Here, we have a custom tool (far right) that lets you quickly construct a right triangle by simply plotting 2 points and THEN entering the measure of one of its acute interior angles. [br][br][b]Note: [/b][br]If you select the RightTriangle tool (far right), simply plot 2 points. Then, enter in the measure of any acute angle. (You can also, if you choose, enter [math]\alpha[/math] = name of slider) if you wish to quickly change the size of this acute angle.
Quick (Silent) Demo: How to Use
– Tim Brzezinski

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4.

Even and Odd Functions

  • 1. Sine and Cosecant Functions (Special Property)
  • 2. Sine Identity
  • 3. Animation 150
  • 4. Cosine and Secant Functions (Special Property)
  • 5. Cosine Identity
  • 6. Animation 149
  • 7. Tangent and Cotangent Functions (Special Property)
  • 8. Animation 145
  • 9. Tangent Identity
  • 10. Animation 151

4.1.

Sine and Cosecant Functions (Special Property)

Suppose [math]\theta[/math] is an angle drawn in standard position. [color=#666666][b]Let [i]P[/i]([i]x[/i], [i]y[/i]) be any point in the coordinate plane[/b][/color] and let[color=#666666][b] [i]r[/i] = the distance from [i]P[/i] to the origin[/b][/color]. [br][br]Recall [math]sin\left(\theta\right)=\frac{y}{r}[/math] and [math]csc\left(\theta\right)=\frac{r}{y}[/math]. [br][br]Interact with the applet below for a minute or two. Then answer the questions that follow. [br][color=#666666][b](Be sure to move point [i]P[/i] to various locations!) [/b][/color][br][br]
1.
Regardless of where [i][color=#666666][b]P[/b][/color][/i] lies, what is the relationship between the values of the ratios [math]sin\left(-\theta\right)[/math] and [math]sin\left(\theta\right)[/math]?
2.
Regardless of where [i][color=#666666][b]P[/b][/color][/i] lies, what is the relationship between the values of the ratios [math]csc\left(-\theta\right)[/math] and [math]csc\left(\theta\right)[/math]?
3.
What do these 2 observations imply about the sine and cosecant functions? (Click [url=https://www.geogebra.org/m/pb8Drtd5]here[/url] and/or [url=https://www.geogebra.org/m/GY9tNvfB]here[/url] for a hint!)
– Tim Brzezinski

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5.

Graphing and Writing Trig Functions

  • 1. Unit Circle to Sine and Cosine Functions
  • 2. Sine & Cosine Period Action (1)!
  • 3. Animation 190
  • 4. Animation 191
  • 5. Sine & Cosine Period Action (2)!
  • 6. Tangent Function Exploration
  • 7. Tangent Period Action (1)!
  • 8. Tangent Period Action (2)!
  • 9. Animation 192
  • 10. Cotangent Function Exploration
  • 11. Transforming Sine and Cosine Functions (1): Dynamic Illustrator
  • 12. Transforming Sine and Cosine Functions (2): Dynamic Illustrator
  • 13. Transforming the Sine Function (All Transformations)
  • 14. Graphing Sine & Cosine Functions (II)
  • 15. Graphing Sine & Cosine Functions (I)
  • 16. Transformations of a Sinusoid
  • 17. Quiz: Graphing Sine & Cosine Functions (Amplitude & Vertical Shift Only)
  • 18. Writing Sine and Cosine Functions (with Amplitude & Vertical Shift Changes)
  • 19. Sine Function Modeling (y = a sin(x) + d) Quiz
  • 20. Cosine Function Modeling (y = a cos(x) + d) Quiz
  • 21. Modeling with Trigonometric Functions
  • 22. Sine & Cosine Function Modeling: Finding Correct Parameters
  • 23. Ferris Wheel Action!!!

5.1.

Unit Circle to Sine and Cosine Functions

Creation of this resource was inspired by [url=https://www.geogebra.org/m/S2gMrkbD]this resource[/url] and [url=https://www.geogebra.org/m/MjFgAfBv]this resource[/url] created by [url=https://www.geogebra.org/u/orchiming]Anthony Or[/url]. [br][br]Slide the [math]\theta[/math] slider first. Explore!
– Tim Brzezinski

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6.

Inverse Trig Functions

  • 1. Inverse Relations: Graphs
  • 2. Sine Function Domain Restriction Options?
  • 3. Cosine Function Domain Restriction Options?
  • 4. Tangent Function Domain Restriction Options?
  • 5. Inverse Trig Functions: Intuitive Explorations

6.1.

Inverse Relations: Graphs

[color=#000000]Recall that, for any relation, the graph of this relation's inverse can be formed by reflecting the graph of this relation about the line y = x. [br][br]Recall that all functions are relations, but not all relations are functions. [br]Again, what causes a relation to be a function? Explain. [br][br]In the applet below, you can input any function [i]f[/i] and restrict its natural domain, if you choose, to input (x) values between -10 and 10. You also have the option to graph the function over its natural domain. [br][br]Interact with this applet for a few minutes, then complete the activity questions that follow. [/color]
[color=#000000][b]Directions: [/b][br][br]1) Choose the [b]"Default to Natural Domain of f"[/b] option. [br]2) Enter in the [/color][color=#980000][b]original function[/b][/color][math]f\left(x\right)=0.2x^2[/math][color=#000000]. [br]3) Choose [/color][color=#38761d][b]"Show Inverse Relation"[/b]. [/color][br][color=#000000]4) Is the [/color][color=#38761d][b]graph of this inverse relation[/b][/color][color=#000000] the graph of a function? Explain why or why not. [br]5) If your answer to (4) above was "no", uncheck the [b]"Default to Natural Domain of f"[/b] checkbox. [br]6) Now, can you come up with a set of Xmin and Xmax values so that the function shown has an inverse [br] that is a function? Explain. [br][br]At any point in this investigation, do the following:[br][br]Use the [b]Point On Object[/b] tool to plot a point on the original function.[br]Then, use the [b]Reflect About Line[/b] tool to reflect this point about the line y = x. [br]What do you notice about the coordinates of this point's reflection? Where does this point lie? [br][br]Repeat steps (1) - (6) again, this time for different functions [i]f[/i] provided to you by your instructor. [/color]
– Tim Brzezinski

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7.

Solving Trig Equations and Modeling

  • 1. Ferris Wheel (1): Modeling with Trigonometric Functions
  • 2. Ferris Wheel (2): Modeling with Trigonometric Functions

7.1.

Ferris Wheel (1): Modeling with Trigonometric Functions

This applet graphs the height of an person riding a Ferris Wheel vs. time. [br]Use this applet as a resource to check solutions to problems involving this context. [br][br][b]There are many parameters you can adjust here: [/b][br][br]Period[br]Number of Revs to Complete[br][color=#980000]Height of Lowest Car[/color][br]Diameter of Wheel[br][br]You can also manually enter the [b][color=#ff0000][i]x[/i]-coordinate of the red point[/color][/b] and/or the [b][color=#9900ff][i]y[/i]-coordinate of the purple point.[/color][/b] (These points are also moveable.) [br]
Quick Demo. (BGM: Simeon Smith)
– Tim Brzezinski

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8.

Proving Trig Identities

  • 1. New Trig IDs From Similar Right Triangles
  • 2. New Trig ID's from Similar Right Triangles (V2)
  • 3. Pythagorean Trigonometric Identity (1)
  • 4. Trig ID Movie (II)
  • 5. Trig ID Movie (III)

8.1.

New Trig IDs From Similar Right Triangles

Recall the definitions of the 6 trigonometric functions defined at an angle drawn in standard position within the coordinate plane. (These ratios were defined in terms of [i]x[/i], [i]y, [/i]& [i]r[/i]). [br][br]Interact with this diagram for a minute or two. (The 2 LARGE POINTS are moveable). [br]Then, answer the question prompts that follow.
1.
Explain why each segment IS what it is. (Some are much easier than others). [br][br]For example, how do we know the [b][color=#9900ff]purple segment has a length = to the tangent of[/color][/b] [math]\theta[/math]?
2.
How many pairs of similar triangles do you see here? How do we know these triangles you reference are all similar to each other?
3.
You have previously learned that similar triangles have corresponding sides that are in proportion. That is, ratios of corresponding sides of similar triangles are all equal in value. [br][br][i]Given this fact, try to author other trig identities from this picture.[/i] You can type them in the space below. Or, even better, feel free to type or use the digital pen to write them in the app below this space. [br]
Write some new trig identities here!
Write some new trig identities here! Use this space if you run out of room above.
– Steve Phelps, Tim Brzezinski

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9.

Proving Advanced Trig Identities

  • 1. Sine & Cosine of a Sum: Discovery
  • 2. Trig ID: Geometric Investigation
  • 3. Tangent of a Sum: Discovery
  • 4. Tangent of a Difference: Trig Identity Puzzle + Proof!
  • 5. Cotangent of a Sum: Discovery
  • 6. Bizzare Trig Identity?
  • 7. Animation 105
  • 8. sin(75) & cos(75) activity

9.1.

Sine & Cosine of a Sum: Discovery

[color=#000000]There are 8 expressions shown off to the right.  [/color]Your job is to carefully drag each expression (label) next to the segment (in the figure) whose length is given by this expression.  [br][br]After doing so, please answer the 2 questions that follow.
Given what you see in your diagram, write an equivalent expression for [math]\sin\left(\alpha+\beta\right)[/math].
Given what you see in your diagram, write an equivalent expression for [math]\cos\left(\alpha+\beta\right)[/math].
– Tim Brzezinski

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10.

Other Functions Resources

  • 1. Functions Resources

10.1.

Functions Resources

[list][*][b][url=https://www.geogebra.org/m/k6Dvu9f3]Interpreting Functions[/url][/b][/*][*][b][url=https://www.geogebra.org/m/uTddJKRC]Building Functions[/url][/b][/*][*][b][url=https://www.geogebra.org/m/GMvvpwrm]Linear, Quadratic, and Exponential Functions[/url][/b][/*][*][b][url=https://www.geogebra.org/m/aWuJMDas]Trigonometric Functions[/url][/b][/*][/list]
Half-life function: Quick Exploration. (Large point & slider moveable.)
What does it mean for a function to be odd? (Points moveable.)
– GeoGebra Team, Tim Brzezinski

Information