THE "THREE BUTTERFLIES" THAT LIVE WITHIN THE UNIT CIRCLE

By: Vernon Willie, Asst. Prof.

San Juan College

4601 College Blvd.

Farmington, NM.

(10/09/02)

 

 

 

 

 

 

Table of Contents

  • Nitshakees
    • Observation
  • Nahat'l
    • Trignometric Terms
    • Angles
    • Unit Circle
    • Special Angles
  • Iin
    • "Short" Butterfly
    • "Medium" Butterfly
    • "Skinny" Butterfly
  • Coordinate Points
    • "Short" Butterfly
    • "Medium" Butterfly
    • "Skinny" Butterfly
    • Intervals of on the Unit Circle
  • Sihasin
    • Questions
    • Assurance

  • 2
    • 2
  • 2
    • 2
    • 2
    • 4
    • 5
  • 5
    • 6
    • 6
    • 7
  • 7
    • 7
    • 8
    • 8
    • 9
  • 11
    • 11
    • 11

 

 

All rights reserved. No part of this module may be reproduced, in any form or by any means, without the permission in writing from the author.

Page 1

Nitshakees - 2

Mathematics and Navajo Culture are considered to be two different worlds. But, as you begin to understand and get a concept of both philosophies, we see a few of the similarities. The most intriguing similarity that I find is that both mathematics and Navajo culture immolate from all aspects of nature.  In this module, we look at mathematics with a little touch of nature; we call upon the "butterfly" to assist us in our understanding of radian measure angles in the subject of Trigonometry. Below are several questions that are related to the radian measures and the unit circle.

What are the Radian Measures?

What is the Unit Circle?

Why are there so many radian measures to "memorize"?

Why can't we just use Degree Measure instead of Radian Measure?

Is it possible to create patterns out of the radian measures on the unit circle?

 

 

We hope to answer these and any other questions that may bring us curiosity. Thus, our focus will be on the three butterflies and their relationship to radian values, coordinate points and any radian correspondence.

Trigonometry and Angles - 2

Before we are to reach and discuss the forms of the "butterflies", it is always good to consider a little bit of a review.
The module is written with the understanding that the reader either is currently taking trigonometry or has taken it in the past. We will begin with a short review from the beginning, what we define as, "Trigonometry".

In mathematics, Trigonometry has been viewed as the study of the triangle; it was developed to find the "measurements of parts" (sides and angles) of a triangle when other parts of the triangle were known. To get to the topic of the module, it does not hurt to do a little bit of a review.
Recall that an Angle is created when we rotate what is called a Half-line (a.k.a., Ray) about its end point, that is called the "Vertex". There must exist an "initial position" (called the Initial Side) of the half-line and as we rotate the half-line, we create a "terminal position" (called the Terminal Side) of the half-line.

Let us denote the "initial side" of half-line as, OA and the "terminal side" as, OB.  We will also use the symbol to denote that we are referring to an angle. With all in place we can actually see a visual description of AOB below.

Page 2

Standard Position - 3

If the rotation of the "terminal side" is counterclockwise, we consider the angle to be "positive". If the rotation of the "terminal side" is clockwise, we consider the angle to be "negative".

In the Navajo culture, "clockwise procedure" is considered to be a "positive" way of life. For example, when one enters a Hogan, it is encouraged that as you enter, you proceed to the left and continue around the Hogan in a "clockwise" fashion. Some consider this a blessing upon yourself through your life.

It was considered long ago that the concept of an angle should be given a "home". Thus, it is placed on the Cartesian coordinate system in the following manner. The "vertex" is positioned directly at the origin and the "initial side" of the angle is placed directly on the positive x-axis. We can then say that the angle is in Standard Position.

 

Radian Measure - 3

Recall that there are two different forms of measurements used when measuring angles. The most common and well appreciated is the form of Degree Measure.

The other form of measurement is usually not so well appreciated and sometimes considered a little more complex to understand is the form of Radian Measure. In this module, we will look at the form of radian measure to get a better grasp of the concept.

In order to look at radian measure, we must first consider a circle with a radius of length r, and the center of the circle located at the origin of the Cartesian coordinate system. We begin in standard position, where the initial side and terminal side are both "lying" on the positive x-axis. As we rotate the terminal side counterclockwise, we begin to create an angle. We continue to rotate the terminal side until we create an arc  of length, r. We will call the new angle that we created, .

Page 3

 

Unit Circle - 4

Notice that the angle is opposite the arc AB. The arc AB is of length, r, thus  is said to be a measure of 1 radian.  Now keep in mind, the angle measurement of "1 radian" is completely independent of the size of the circle.

For example, if you were to have a circle that has a radius length of 10 ft. and you measured out an arc length of 10 ft. on the circle that is still considered being "1 radian" measure.

Now we are ready for the Unit Circle. The Unit Circle is a circle that is centered at the origin of the Cartesian coordinate system and has a radius of length 1 unit.

 

Notice that when rotates counterclockwise, we create a quarter rotation of a circle and stop on the positive y-axis. The radian angle measure is called, "/ 2". We claim,
"= / 2".

As we continue to make a counterclockise rotation with and stop at the negative x-axis, the radian angle measure is called, . So now we may indicate, = .  As we continue the counter clockwise rotation and stops on the negative y-axis, the radian angle measure is called,"  ".  Now we can claim that   Continuing in the counterclockwise rotation and return back to the positive x-axis, has "come full-circle" or made one complete revolution. The radian measure is now "2". Thus we conclude that= 2.

When the terminal side of an angle coincides on one of the coordinate axes, we recognize the angle as a "Quadrantal angle". We can recognize the above angles,    as quadrantal angles.

 

Page 4

Special Angles - 5

As we continue to learn about angles of radian measure angles on the Unit Circle, we see that there are quite a few angles.

The "special angles" that we are introduced to are All of a sudden there are an additional "9" other angles or radian measure that correspond to the special angles.  We are now at a total of "16" angles of radian measure that are on the unit circle.

Iin: - 5

One of the challenges in Trigonometry is learning the actual location of each of the special angles. One method that has assisted me (and better than "memorizing") in actually learning the angles of radian measure is using what I call, "The Three Butterflies".

Imagine three butterflies that live inside the unit circle. They are the "Short" butterfly, the "Medium butterfly" and the "Skinny" butterfly. We look at the "wingtips" of the butterflies and how they correspond to the radian measures.  We also look at the coordinate points that they give based on their location on the unit circle. We also see that the radian measure for correspond with each other.

Page 5

 

Short Butterfly - 6

We will start with the "Short Butterfly " that you see below. Each of the corner "wingtips" has a radian measure. Also notice that the radian measures all have a denominator value of "6".

 

 

Medium Butterfly - 6

The "Medium Butterfly" will have radian measures with the denominator value of "4". Notice the corner of its "wingtips" below.

 

 

Page 6

Skinny Butterfly -7

The "Skinny Butterfly" will have radian measures with the denominator value of "3". Notice the corner of its "wingtips" below.

 

 

Coordinate Points - 7

If you were to look at each butterfly separately, you will notice that the coordinate points, that are located at the "wingtips", all have the same absolute values. It is only due to the location of which quadrant each "wingtip" is in that the values will be either positive or negative. See the butterflies below.

"Short Butterfly" - 7:  

 

 

Page 7

Medium Butterfly - 8

"Medium Butterfly":

 

 

"Skinny Butterfly" - 8:

 

Intervals of

Page 8

 On the Unit Circle - 9

When we look at the unit circle as intervals of the "special angles"  between , we notice that  has a total of "12" intervals.  Four angles will correspond with the angles of , and "4" angles will correspond with the "Quadrantal angles", when simplified.  If you look a little closer, one can see both the "Small Butterfly" and the "Skinny Butterfly".

 

 

Page 9

 

***Notice:

 

 

The radian angle of  will have a total of "8" angles. Four of the angles will correspond with the "Quadrantal angles", when simplified. Notice that one can visualize the "Medium Butterfly".

 

 

***Notice:

 

 

 

The radian angle of  will have a total of "8" angles also. Only two of the angles will correspond with the "Quadrantal angles", when simplified. Those angles are on the x-axis. Also, one can see the "Skinny Butterfly".

 

Page 10

 

***Notice:

 
 

**Sihasin: - 11**

As we look back on the module, we started out with a short comment on mathematics and Navajo culture.

- Next, we followed with a review of the mathematics subject; Trigonometry. The review was to prepare us for the view or interpretation of the "Three Butterflies".

- And we finally got to meet the "butterflies" and to attempt to get an understanding of what they had to offer in terms of mathematics. As we reflect on the module, we want to ask ourselves several questions.

- Below is a list of several questions to get started but also questions that you may have but are not listed.

  • Do you think that this was an interesting way to interpret mathematics?

  • What mathematical concepts were helpful in this study?

  • Are there any other questions that you had that were answered by this model?

  • What other creature "representation" would you have used instead of a "butterfly"?

 

Page 11