Trigonometry

Arthur Lee, Oct 27, 2014

Trigonometric Ratios and Functions This is my collection of trigonometry materials, from junior to senior levels. Some items are copied or adapted from the [url=https://www.gmath.hk]gMath site[/url]. There are also Desmos activities with similar design in my [url=https://teacher.desmos.com/collection/5caef6421e821b51df80373f]collection page[/url], especially the core idea of using and emphasising the unit circle in the process of learning.

Table of Contents

  1. Right Triangles
    1. Introduction to Sine
    2. Trigonometric Ratios: Sine 三角比：正弦
    3. Introduction to Cosine & Tangent
    4. Trigonometric Ratios: Cosine & Tangent 三角比：餘弦和正切
    5. Simple Sine Table
    6. Sine and Cosine Table
    7. recognise trigonometric ratios
    8. Quiz on Trigonometric Ratios 辨認三角比：測驗
    9. Regiomontanus
    10. trig ratio
    11. lighthouse
  2. All Triangles
    1. Sine Law
    2. Cosine Law
    3. Sin Law (basic)
    4. Proof without words: Cosine Rule (餘弦公式)
  3. Unit Circle
    1. unwind to understand radian
    2. 任意角的三角比
    3. trigonometric ratios for all angles
    4. rectangular vs polar (1)
    5. rectangular vs polar (2a)
    6. rectangular vs polar (2b)
    7. unit circle and symmetry
    8. Unit Circle & Symmetry (Practice)
    9. unit circle pattern 1
    10. unit circle pattern 2
    11. parametric
    12. unit circle pattern example
    13. polar and spiral
    14. Polar and Rectangular
    15. Introduction to Polar Coordinates 極坐標簡介
    16. 極坐標測驗
  4. making patterns and animation
    1. Sinimation
    2. sine curve rotated
    3. Cosine Shift
    4. Dot Pattern 1
    5. Dot Pattern 2
    6. Dot Pattern 3
    7. Happy Halloween
    8. Halloween with sound
    9. sun and sea
    10. Moon and Sea
    11. creating symmetry

1.

Right Triangles

introduce trigonometric ratios in Key Stage 3 (junior secondary level), focusing on sides and ratios in right triangles

  • 1. Introduction to Sine
  • 2. Trigonometric Ratios: Sine 三角比：正弦
  • 3. Introduction to Cosine & Tangent
  • 4. Trigonometric Ratios: Cosine & Tangent 三角比：餘弦和正切
  • 5. Simple Sine Table
  • 6. Sine and Cosine Table
  • 7. recognise trigonometric ratios
  • 8. Quiz on Trigonometric Ratios 辨認三角比：測驗
  • 9. Regiomontanus
  • 10. trig ratio
  • 11. lighthouse

1.1.

Introduction to Sine

– GeoGebra Institute of Hong Kong, Alex CHIK

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1.6.

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1.7.

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1.9.

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1.10.

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1.11.

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2.

All Triangles

  • 1. Sine Law
  • 2. Cosine Law
  • 3. Sin Law (basic)
  • 4. Proof without words: Cosine Rule (餘弦公式)

2.1.

Sine Law

– Arthur Lee

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3.

Unit Circle

In this section, we focus on the basic definition of sine and cosine ratio/function according to the unit circle. In doing so, some activities can help learners to understand the meaning of these trig values based on patterns related to circles. Knowledge of polar coordinates could be relevant and useful. Although we need not use such notation explicitly, describing position in terms of distance and direction could be a good purpose for using trig ratios. As a starting point, we can simply consider cosine and sine ratios as coordinates of points on a unit circle. Based on this, we can easily talk about different kinds of circular patterns. [url=https://www.evernote.com/shard/s3/sh/e5ba1be0-ab66-1287-b448-a947a12a5b70/9c99c559a28e83e22afea24681ed21ac]Know more about this idea in Sawyer's book. [/url]

  • 1. unwind to understand radian
  • 2. 任意角的三角比
  • 3. trigonometric ratios for all angles
  • 4. rectangular vs polar (1)
  • 5. rectangular vs polar (2a)
  • 6. rectangular vs polar (2b)
  • 7. unit circle and symmetry
  • 8. Unit Circle & Symmetry (Practice)
  • 9. unit circle pattern 1
  • 10. unit circle pattern 2
  • 11. parametric
  • 12. unit circle pattern example
  • 13. polar and spiral
  • 14. Polar and Rectangular
  • 15. Introduction to Polar Coordinates 極坐標簡介
  • 16. 極坐標測驗

3.1.

unwind to understand radian

You may use this applet to study the meaning of radian measure. The highlighted arc can be unwinded to match a straight vertical segment to facilitate the comparison with the radius of the circle. [br][br]Vary the angle/arc or radius to to get a sense of the radian measure.
This is a quick silent demo.
– Arthur Lee

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3.7.

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3.9.

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3.10.

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3.11.

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3.12.

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3.13.

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3.14.

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4.

making patterns and animation

create patterns, animation, drawing using trigonometric and other functions/graphs

  • 1. Sinimation
  • 2. sine curve rotated
  • 3. Cosine Shift
  • 4. Dot Pattern 1
  • 5. Dot Pattern 2
  • 6. Dot Pattern 3
  • 7. Happy Halloween
  • 8. Halloween with sound
  • 9. sun and sea
  • 10. Moon and Sea
  • 11. creating symmetry

4.1.

Sinimation

Sinimation! Have fun with a simple sine curve animated.
Sinimation
Can you recognise the sine graph and parameters making this pattern? [br]Pause the animation. Drag a and b to experiment or look for an image you like.
– Arthur Lee

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4.6.

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4.9.

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4.10.

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4.11.

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