Trigonometry Maths Made Easy

Tim Brzezinski, Maths Made Easy Suresh, Feb 26, 2020

CBSE Grade 10,11,12

Table of Contents

  1. Angles in Standard Position (and Related Vocabulary)
    1. Angles in Standard Position
    2. Angle Orientations (Standard Position)
    3. Coterminal Angles Action!!!
  2. The 6 Trigonometric Ratios; Right Triangle Trigonometry; Applications
    1. Finding Exact Trig Ratios: Quiz Question Generator
    2. Evaluating Trig Functions Given a Point on the Terminal Ray
    3. Trig Ratio Possibilities (Hints)
    4. Right Triangle Generator for Right Triangle Trigonometry
    5. Right Triangle Trigonometry: Intro
    6. True Meaning of Sine, Cosine, Tangent Ratios within Right Triangles
    7. What "CO" in COsine Means
  3. 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)
  4. Introduction to Radians; Arc Length and Area of a Sector
    1. Circle vs. Sphere
    2. Circle Terminology
    3. TRUE MEANING of π
    4. 1 Radian: Clear Definition
    5. Radian Illustrator
    6. DEG to RAD and RAD to DEG Illustrator
    7. Arc length: Quick Investigation
    8. Angular to Linear: Without Words
  5. 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. Writing Sine and Cosine Functions (with Amplitude & Vertical Shift Changes)
    7. Transforming the Sine Function (All Transformations)
    8. Sine & Cosine Period Action (1)!
    9. Graphing Sine & Cosine Functions (I)
    10. Graphing Sine & Cosine Functions (II)
    11. Inverse Trig Functions: Intuitive Explorations
  6. 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. Vectors
    1. Vectors: Magnitude & Direction (I)
    2. Scaling Vectors
    3. Vector Addition
    4. Quiz: Adding Vectors Geometrically
    5. Vector Subtraction

1.

Angles in Standard Position (and Related Vocabulary)

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

1.1.

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

1.2.

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1.3.

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2.

The 6 Trigonometric Ratios; Right Triangle Trigonometry; Applications

  • 1. Finding Exact Trig Ratios: Quiz Question Generator
  • 2. Evaluating Trig Functions Given a Point on the Terminal Ray
  • 3. Trig Ratio Possibilities (Hints)
  • 4. Right Triangle Generator for Right Triangle Trigonometry
  • 5. Right Triangle Trigonometry: Intro
  • 6. True Meaning of Sine, Cosine, Tangent Ratios within Right Triangles
  • 7. What "CO" in COsine Means

2.1.

Finding Exact Trig Ratios: Quiz Question Generator

– GeoGebra Team, Tim Brzezinski

2.2.

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2.3.

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2.6.

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2.7.

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3.

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)

3.1.

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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3.3.

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3.5.

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3.6.

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3.7.

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4.

Introduction to Radians; Arc Length and Area of a Sector

  • 1. Circle vs. Sphere
  • 2. Circle Terminology
  • 3. TRUE MEANING of π
  • 4. 1 Radian: Clear Definition
  • 5. Radian Illustrator
  • 6. DEG to RAD and RAD to DEG Illustrator
  • 7. Arc length: Quick Investigation
  • 8. Angular to Linear: Without Words

4.1.

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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4.4.

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4.5.

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4.6.

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4.7.

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4.8.

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5.

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. Writing Sine and Cosine Functions (with Amplitude & Vertical Shift Changes)
  • 7. Transforming the Sine Function (All Transformations)
  • 8. Sine & Cosine Period Action (1)!
  • 9. Graphing Sine & Cosine Functions (I)
  • 10. Graphing Sine & Cosine Functions (II)
  • 11. Inverse Trig Functions: Intuitive Explorations

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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5.4.

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5.5.

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5.6.

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5.7.

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5.8.

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5.9.

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5.11.

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6.

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)

6.1.

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

6.2.

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6.3.

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7.

Vectors

Applets about Vector Operations

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

7.1.

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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