Table of Contents
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. 
Nits_{}hakees  2Mathematics 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.
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  2Before 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 Halfline (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 halfline and as we rotate the halfline, we create a "terminal position" (called the Terminal Side) of the halfline.Let us denote the "initial side" of halfline 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. 
Standard Position  3

Radian Measure  3

Unit Circle  4






Medium Butterfly  8



***Notice:


***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 xaxis. 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.
