Nitsáhákees:
The Navajos are a Native American tribe that lives in the southwestern part of the United States. Their reservation expands into three states from northwestern New Mexico, northeastern Arizona and southeastern Utah. The Navajos live in the boundaries of what is considered the four sacred mountains. They are as follows below:
1.Mountain to the East: (Sisnaajiní) Blanca Peak, near Alamosa, CO.
2.Mountain to the South: (Tsoodzil) Mount Taylor, near Grants, NM.
3.Mountain to the West: (Dook’o’oosliid) Francisco Peaks, near Flagstaff, AZ.
4.Mountain to the North: (Dibé Nitsaa) Hesperus Peak, near Hesperus, CO.
There is also a Center mountain, it is called, Huerfano Mountain (Dzilná’oodilii), near Bloomfield, NM.
Below is a visual description of where the mountains are located; we will use the center mountain as our origin.

With the center mountain as our origin, we are able to get an idea of where the four directional mountains are located.
Nahatá:

Angle Measurements

There are several different methods used in mathematics and engineering/science fields to look at angle measures. We will take a look at several, the first will be Navigational Direction Reading followed by the Traditional Trigonometric Angle Reading and the last will be a new form which I will call, "Angle Reading with Navajo Influence".

Navigational Reading

One method that we will use is the Navigational Direction. Navigational direction is used, for example in naval navigation or in aerial navigation. The directions to a certain location are given as an angle that is rotated in the clockwise direction from the north. Given the reading of  , we are given the direction of north. Given the reading of   , we are given the direction of due east. Given the reading of     , we are given the direction of due south. Given the reading of     , we are given the direction of due west. Given the reading of      , we are given the direction of due northwest. Below is a list of the navigational readings for each of the directional mountains:

Navigational Readings

 

From\To: Blanca Peak Mt. Taylor San Francisco Pk. Hesperus Peak
Huerfano Mtn.

 

Blanca Peak:
Mount Taylor:
San Francisco Peaks:
Hesperus Peak:

The northern axis (also known as the positive y-axis) is considered our initial side. Each line that extends from the center mountain to each directional mountain is our terminal side. Thus we are able to create the navigational readings. Below is also a list of the radian measures for each degree angle.
From\To: Blanca Peak Mt. Taylor San Francisco Pk. Hesperus Peak
Huerfano Mtn.
1.0122 rads. 2.9792 rads. 4.3982 rads. 6.0946 rads.

 

Traditional Trigonometric Angle Reading

 

Let us now look at the angle measurements as we would in a traditional trigonometry course. Recall that when we place two half lines and its vertex in standard position, the vertex would be at the origin and the initial side of the angle would be on the positive x-axis. If we rotate an angle counterclockwise, then we consider the angle to have a positive measure. Thus, we get the following when we look at each mountain separately.
Blanca Peak:
Mount Taylor:
San Francisco Peaks:
Hesperus Peak:
From\To: Blanca Peak Mt. Taylor San Francisco Pk. Hesperus Peak
Huerfano Mtn.
0.5585 rads. 1.7559 rads. 3.4557 rads. 4.8747 rads.

 

Viewing the four sacred mountains of the Navajos from a “traditional trigonometric” standpoint can be a little unusual and maybe even uncomfortable for some. As mentioned earlier, in trigonometry, when we rotate an angle counterclockwise, we create a "positive" angle measure. In the Navajo culture, it is taught that the journey through life is circular. And the concept of "positive-ness" corresponds with a "clockwise" motion. For example, when one enters a Hogan (a Navajo dwelling), it is considered respectful to go around the Hogan in the "clockwise" (positive) direction. When recognizing the four sacred mountains, the recognition is in a "clockwise" fashion; begin in the East, and then the South, followed by the West, concluded in the North and returned back to the East.
I do realize that I recognized the mountains in a counterclockwise fashion to recognize the angle measurements for each mountain. I do this out of good faith, to compare that there are similarities and differences in looking at angles from a western philosophical and traditional Navajo philosophical point of view. In no way whatsoever do I mean any disrespect to the mountains and the Navajo teachings.
Iiná:

Angle Reading with a Navajo Influence

To bring in a sense of balance we will now look at the angle measurements from a Navajo philosophical point of view. Again, as mentioned earlier, we consider "positive-ness" in the clockwise direction. Thus, below is a view of the mountains starting at standard position (positive x-axis). Since Blanca Peak is the mountain to the East, we will consider the line that connects the center mountain to the east mountain as our standard position…our "Navajo East". Now by rotating the angle clockwise, we recognize each mountain in the clockwise direction.
Blanca Peak:
Mount Taylor:
Humphreys Peak:
Hesperus Peak
Below is a table that displays the angle measurements with a little bit of a Navajo influence; let the line that connects Huerfano Mountain and Blanca Peak be our standard position as we begin to rotate angles in the clockwise direction.
From\To: Blanca Peak Mt. Taylor San Francisco Pk. Hesperus Peak
Huerfano Mtn.
0 rads. 1.9669 rads. 3.3859 rads. 5.0824 rads.

 

Sihasin:
We have taken a look at the four sacred mountains of the Navajos (and the center mountain) from several different viewpoints. The first involved a method of Trigonometry from an application viewpoint. The second involved the more pure or traditional Trigonometric methods in reference to standard position. And the third involved a little big of a Navajo Influence. As we reflect, several questions come to mind: