Trigonometry

Tim Brzezinski, Ronan Downes, 4 Mar 2020

Accompanies MATH 115: Trigonometry (CCSU) course.

Table of Contents

  1. Preliminary Content
    1. Perpendicular Lines: Quick Intro
    2. Complementary Meaning?
    3. Supplementary Angles (Quick Exploration)
    4. Isosceles Triangle (Quick Construction Technique) V1
    5. Equilateral Triangle Construction (Dynamic Illustration)
    6. Proof Without Words
    7. Proof Without Words
    8. Practice: Pythagorean Theorem (1)
    9. Quiz: Pythagorean Theorem (2A)
    10. Quiz: Pythagorean Theorem (2B)
    11. Simplifying Radicals (I)
    12. Simplifying Radicals (II)
    13. Triangle Angle Sum Theorem (V4)
    14. Animation 171
    15. Isosceles Right Triangle: Quick Investigation
    16. Another Special Right Triangle (Guided Discovery)
    17. Special Right Triangle (II)
    18. Vertical Angles Exploration (1)
    19. Naming Angle Positions
    20. Transversal Intersects Parallel Lines
    21. Corresponding Angles: Quick Investigation
    22. Alternate Interior Angles: Quick Investigation
    23. Alternate Exterior Angles: Quick Investigation
    24. Same-Side-Interior Angles: Quick Investigation
    25. Same-Side-Exterior Angles: Quick Investigation
    26. Similar Figures: Dynamic Illustration
    27. AA Similarity Theorem
    28. SSS ~ Theorem (V2)
  2. Angles in Standard Position (and Related Vocabulary)
    1. Angles in Standard Position
    2. Angle Orientations (Standard Position)
    3. Coterminal Angles Action!!!
  3. The 6 Trigonometric Ratios; Right Triangle Trigonometry; Applications
    1. Right Triangles: Identifying Sides WRT Acute Angles
    2. Identifying Trig Ratios: Quick Formative Assessment
    3. Finding Exact Trig Ratios: Quiz Question Generator
    4. Quiz: Evaluating Trigonometric Functions of Angles Given a Point on its Terminal Ray
    5. Evaluating Trig Functions Given a Point on the Terminal Ray
    6. Quiz: Quadrant Logic (Trig)
    7. Trig Quadrant Logic (2)
    8. Trig Ratio Possibilities (Hints)
    9. Quiz: Evaluating Trig FNS of Angles (Quadrant Logic): V1
    10. Quiz: Evaluating Trig FNS of Angles WITH Quadrant Logic (V2)
    11. Right Triangle Generator for Right Triangle Trigonometry
    12. Right Triangle Trigonometry: Intro
    13. True Meaning of Sine, Cosine, Tangent Ratios within Right Triangles
    14. Trig Ratio Estimations (Right Triangle Context)
    15. Trig Ratio Estimations (Quiz: V1)
    16. Right Triangle Solver: Formative Assessment
    17. Right Triangle Trig: Solving for Sides
    18. Right Triangle Trig: Solving for Sides (2)
    19. Finding Acute Angles of Right Triangles
    20. Quiz: Expressing Angles Using Inverse Trigonometric Function
    21. What "CO" in COsine Means
  4. Trigonometric Functions of Special Angles
    1. Trig Function Values (30 Degrees)
    2. Trig Function Values (45 Degrees)
    3. Trig Function Values (60 Degrees)
    4. Special Right Triangles: Basic Questions
    5. Quiz: Special Right Triangles
    6. Special Angles in Standard Position: Intro (V2)
    7. Special Angles in Standard Position: Intro (V3)
    8. Quiz 1: Trigonometric Functions of Special Angles
    9. Quiz 2: Trigonometric Functions of Special Angles
  5. Introduction to Radians; Arc Length and Area of a Sector
    1. Circle vs. Sphere
    2. Circle Terminology
    3. TRUE MEANING of π
    4. Circumference = ? (Animation)
    5. Circumference = ? (Animation II)
    6. 1 Radian: Clear Definition
    7. Radian Illustrator
    8. Quiz: Sketching Angles in Standard Position
    9. DEG to RAD and RAD to DEG Illustrator
    10. DEG to RAD and RAD to DEG: Question Generator
    11. Arc length: Quick Investigation
    12. Parallelogram: Area
    13. Triangle Area Action!!! (V2)
    14. Circle Area (By Peeling!)
    15. Arc Length and Area of a Sector (V1)
    16. Arc Length and Area of a Sector (V2)
    17. Area of a Sector
    18. Angular to Linear: Without Words
    19. YOUR Linear Speed?
  6. Graphing and Writing Trigonometric Functions
    1. Unit Circle to Sine and Cosine Functions
    2. Tangent Function Exploration
    3. Cotangent Function Exploration
    4. Transforming Sine and Cosine Functions (1): Dynamic Illustrator
    5. Transforming Sine and Cosine Functions (2): Dynamic Illustrator
    6. Quiz: Graphing Sine & Cosine Functions (Amplitude & Vertical Shift Only)
    7. Writing Sine and Cosine Functions (with Amplitude & Vertical Shift Changes)
    8. Periodic Function Action!
    9. Transforming the Sine Function (All Transformations)
    10. Sine & Cosine Period Action (1)!
    11. Graphing Sine & Cosine Functions (I)
    12. Graphing Sine & Cosine Functions (II)
    13. Modeling with Trigonometric Functions
    14. Inverse Trig Functions: Intuitive Explorations
  7. Proving Trig Identities
    1. Trig ID: Geometric Investigation
    2. New Trig IDs From Similar Right Triangles
    3. New Trig ID's from Similar Right Triangles (V2)
  8. Advanced Topics
    1. Special Trigonometric Identities: Quick Exploration
    2. Law of Sines (& Area)
    3. SSA Theorem?
    4. Ambiguous Case (SSA) Illustrator: All Cases
    5. Law of Cosines: Discovery
  9. Bearings (Applications)
    1. Bearing Illustrator (1)
    2. Bearing: Dynamic Illustration
    3. Bearing Illustrator (2)
    4. Bearings: Dynamic Illustrator
  10. Vectors
    1. Vectors: Magnitude & Direction (I)
    2. Scaling Vectors
    3. Vector Addition
    4. Quiz: Adding Vectors Geometrically
    5. Vector Subtraction

1.

Preliminary Content

  • 1. Perpendicular Lines: Quick Intro
  • 2. Complementary Meaning?
  • 3. Supplementary Angles (Quick Exploration)
  • 4. Isosceles Triangle (Quick Construction Technique) V1
  • 5. Equilateral Triangle Construction (Dynamic Illustration)
  • 6. Proof Without Words
  • 7. Proof Without Words
  • 8. Practice: Pythagorean Theorem (1)
  • 9. Quiz: Pythagorean Theorem (2A)
  • 10. Quiz: Pythagorean Theorem (2B)
  • 11. Simplifying Radicals (I)
  • 12. Simplifying Radicals (II)
  • 13. Triangle Angle Sum Theorem (V4)
  • 14. Animation 171
  • 15. Isosceles Right Triangle: Quick Investigation
  • 16. Another Special Right Triangle (Guided Discovery)
  • 17. Special Right Triangle (II)
  • 18. Vertical Angles Exploration (1)
  • 19. Naming Angle Positions
  • 20. Transversal Intersects Parallel Lines
  • 21. Corresponding Angles: Quick Investigation
  • 22. Alternate Interior Angles: Quick Investigation
  • 23. Alternate Exterior Angles: Quick Investigation
  • 24. Same-Side-Interior Angles: Quick Investigation
  • 25. Same-Side-Exterior Angles: Quick Investigation
  • 26. Similar Figures: Dynamic Illustration
  • 27. AA Similarity Theorem
  • 28. SSS ~ Theorem (V2)

1.1.

Perpendicular Lines: Quick Intro

WARM UP:
The lines below are said to be [b]perpendicular lines[/b]. [br][br]
Move the LARGE POINTS around a bit, then answer the question that appears below.
[b]What does it mean for two lines to be perpendicular? [/b]
– Tim Brzezinski

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2.

Angles in Standard Position (and Related Vocabulary)

  • 1. Angles in Standard Position
  • 2. Angle Orientations (Standard Position)
  • 3. Coterminal Angles Action!!!

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.

The 6 Trigonometric Ratios; Right Triangle Trigonometry; Applications

  • 1. Right Triangles: Identifying Sides WRT Acute Angles
  • 2. Identifying Trig Ratios: Quick Formative Assessment
  • 3. Finding Exact Trig Ratios: Quiz Question Generator
  • 4. Quiz: Evaluating Trigonometric Functions of Angles Given a Point on its Terminal Ray
  • 5. Evaluating Trig Functions Given a Point on the Terminal Ray
  • 6. Quiz: Quadrant Logic (Trig)
  • 7. Trig Quadrant Logic (2)
  • 8. Trig Ratio Possibilities (Hints)
  • 9. Quiz: Evaluating Trig FNS of Angles (Quadrant Logic): V1
  • 10. Quiz: Evaluating Trig FNS of Angles WITH Quadrant Logic (V2)
  • 11. Right Triangle Generator for Right Triangle Trigonometry
  • 12. Right Triangle Trigonometry: Intro
  • 13. True Meaning of Sine, Cosine, Tangent Ratios within Right Triangles
  • 14. Trig Ratio Estimations (Right Triangle Context)
  • 15. Trig Ratio Estimations (Quiz: V1)
  • 16. Right Triangle Solver: Formative Assessment
  • 17. Right Triangle Trig: Solving for Sides
  • 18. Right Triangle Trig: Solving for Sides (2)
  • 19. Finding Acute Angles of Right Triangles
  • 20. Quiz: Expressing Angles Using Inverse Trigonometric Function
  • 21. What "CO" in COsine Means

3.1.

Right Triangles: Identifying Sides WRT Acute Angles

– Tim Brzezinski

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4.

Trigonometric Functions of Special Angles

  • 1. Trig Function Values (30 Degrees)
  • 2. Trig Function Values (45 Degrees)
  • 3. Trig Function Values (60 Degrees)
  • 4. Special Right Triangles: Basic Questions
  • 5. Quiz: Special Right Triangles
  • 6. Special Angles in Standard Position: Intro (V2)
  • 7. Special Angles in Standard Position: Intro (V3)
  • 8. Quiz 1: Trigonometric Functions of Special Angles
  • 9. Quiz 2: Trigonometric Functions of Special Angles

4.1.

Trig Function Values (30 Degrees)

Interact with the applet below for a few minutes. Then answer the questions that follow.
LARGE POINTS are MOVEABLE. Be sure to slide the slider (lower right) slowly. As you do, pay careful attention to what you see here.
1.
Just by merely observing the dynamics of this applet, what would the [color=#9900ff][b]sine of 30 degrees [/b][/color]be?
2.
What would the [color=#9900ff][b]cosecant of 30 degrees[/b][/color] be?
3.
Do the values of [color=#9900ff][b]either of these ratios[/b][/color] depend upon [color=#666666][b]the length of the circle's radius[/b][/color]? Explain why or why not.
– Tim Brzezinski

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5.

Introduction to Radians; Arc Length and Area of a Sector

  • 1. Circle vs. Sphere
  • 2. Circle Terminology
  • 3. TRUE MEANING of π
  • 4. Circumference = ? (Animation)
  • 5. Circumference = ? (Animation II)
  • 6. 1 Radian: Clear Definition
  • 7. Radian Illustrator
  • 8. Quiz: Sketching Angles in Standard Position
  • 9. DEG to RAD and RAD to DEG Illustrator
  • 10. DEG to RAD and RAD to DEG: Question Generator
  • 11. Arc length: Quick Investigation
  • 12. Parallelogram: Area
  • 13. Triangle Area Action!!! (V2)
  • 14. Circle Area (By Peeling!)
  • 15. Arc Length and Area of a Sector (V1)
  • 16. Arc Length and Area of a Sector (V2)
  • 17. Area of a Sector
  • 18. Angular to Linear: Without Words
  • 19. YOUR Linear Speed?

5.1.

Circle vs. Sphere

[color=#000000]Interact with the applet below for a few minutes, then answer the questions that follow. Be sure to move the BIG POINTS around during your investigation! [/color]
[color=#980000][b]Questions:[/b][/color][br][br][color=#000000]1) What do a CIRCLE and SPHERE have in common? Explain.[br][br]2) How are a CIRCLE and SPHERE different? Explain. [br][br]3) Using what you've observed, write a definition for the term CIRCLE and for the term SPHERE. [br] [i]Hint: Be sure to use the words "locus", "points", and "equidistant" in each definition! [/i][/color]
– Tim Brzezinski

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6.

Graphing and Writing Trigonometric Functions

  • 1. Unit Circle to Sine and Cosine Functions
  • 2. Tangent Function Exploration
  • 3. Cotangent Function Exploration
  • 4. Transforming Sine and Cosine Functions (1): Dynamic Illustrator
  • 5. Transforming Sine and Cosine Functions (2): Dynamic Illustrator
  • 6. Quiz: Graphing Sine & Cosine Functions (Amplitude & Vertical Shift Only)
  • 7. Writing Sine and Cosine Functions (with Amplitude & Vertical Shift Changes)
  • 8. Periodic Function Action!
  • 9. Transforming the Sine Function (All Transformations)
  • 10. Sine & Cosine Period Action (1)!
  • 11. Graphing Sine & Cosine Functions (I)
  • 12. Graphing Sine & Cosine Functions (II)
  • 13. Modeling with Trigonometric Functions
  • 14. Inverse Trig Functions: Intuitive Explorations

6.1.

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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7.

Proving Trig Identities

  • 1. Trig ID: Geometric Investigation
  • 2. New Trig IDs From Similar Right Triangles
  • 3. New Trig ID's from Similar Right Triangles (V2)

7.1.

Trig ID: Geometric Investigation

STUDENTS:
Interact with this applet for a few minutes. [br]As you do, be sure to change the location of the [b][color=#ff00ff]LARGE PINK POINT[/color] [/b]a few times. [br]After doing so, please answer the questions that follow. [br][br]Note: In this animation, lines that appear to be parallel ARE parallel.
1.
Express the area of the original rectangle (at the beginning of the animation) in terms of [math]\theta[/math].
2.
What is the area of the rectangle you see at the end of the animation? Express this area in terms of [math]\theta[/math].
3.
What can we conclude about the areas of the rectangle shown at the beginning and end of this animation? How/why do we know this to be true?
4.
Given your results for (1), (2), and (3) above, write a relationship between the expressions (written in terms of [math]\theta[/math]) you wrote in (1) and (2).
– Tim Brzezinski

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8.

Advanced Topics

  • 1. Special Trigonometric Identities: Quick Exploration
  • 2. Law of Sines (& Area)
  • 3. SSA Theorem?
  • 4. Ambiguous Case (SSA) Illustrator: All Cases
  • 5. Law of Cosines: Discovery

8.1.

Special Trigonometric Identities: Quick Exploration

1.
Write "?" in terms of [math]\theta[/math]. What do you get?
2.
In terms of the symbols "%", "$", and/or "!", what is [math]sin\left(\theta\right)[/math]?
3.
In terms of the symbols "%", "$", and/or "!", what is [math]cos\left(?\right)[/math]
4.
Take your results from (1) and (3) to write a new trigonometric identity. What do you get?
5.
In terms of the symbols "%", "$", and/or "!", what is [math]cos\left(\theta\right)[/math]?
6.
In terms of the symbols "%", "$", and/or "!", what is [math]sin\left(?\right)[/math]
7.
Take your results from (1) and (6) to write a new trigonometric identity. What do you get?
– Tim Brzezinski

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9.

Bearings (Applications)

  • 1. Bearing Illustrator (1)
  • 2. Bearing: Dynamic Illustration
  • 3. Bearing Illustrator (2)
  • 4. Bearings: Dynamic Illustrator

9.1.

Bearing Illustrator (1)

In trigonometry, [b][color=#9900ff]a bearing[/color][/b] is a means of describing one point's location with respect to another.[br][br]The applet below illustrates point B's bearing with respect to point [i]A[/i]. [br]Slide the [b][color=#9900ff]BEARING[/color][/b] slider to change the bearing from point [i]A[/i] to point [i]B[/i]. [br]You can also move any of the 3 points around at any time. [br][br]Then, answer the questions that follow. [br][br]
1.
In words, describe what the bearing from point [i]A[/i] to point [i]B [/i]means. Be sure to be detailed in your description!
2.
Can you give another bearing equivalent to a bearing of 120 degrees? Why is this?
3.
If point [i]B[/i] is directly south of [i]A[/i], what is point [i]B[/i]'s bearing from [i]A[/i]?
4.
If point [i]B[/i] is directly west of [i]A[/i], what is point [i]B[/i]'s bearing from [i]A[/i]?
– Tim Brzezinski

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10.

Vectors

Applets about Vector Operations

  • 1. Vectors: Magnitude & Direction (I)
  • 2. Scaling Vectors
  • 3. Vector Addition
  • 4. Quiz: Adding Vectors Geometrically
  • 5. Vector Subtraction

10.1.

Vectors: Magnitude & Direction (I)

[color=#000000]A vector is a mathematical quantity with both magnitude and direction. [br][br]The[b] magnitude of vector u[/b], denoted |[b]u[/b]|, is it's length. [br][color=#a64d79]One way to express its direction is to give the angle it makes with a horizontal ray (that points to the right) that is parallel to the positive x-axis. This is called the vector's [b]direction angle. [/b][/color][b][br][/b][br]See below. You can also change the locations of the vector's initial point and terminal point[/color]. [br][br][color=#0000ff][b]KEY QUESTION:[/b] Given a vector's magnitude and direction, how can you rewrite this vector in component form? < , > ? [/color]
– Tim Brzezinski

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Information