I/D#1: Unit N Concept 7: The Unit Circle- How do Special Right Triangles and the Unit Circle relate?

What are special right triangles?

• When talking about special right triangles in this class, we are referring to right triangles with specific angles



http://gmatprepster.com/wp-content/uploads/2012/07/GMAT-Right-Triangles-1.png
  

• These are triangles with 30-60-90 and 45-45-90 degree relations and as demonstrated above, the degrees will correlate with the side measures of the triangle with the specified formulas (note that the 30 and 60 degrees can switch spots in the 30-60-90; the side values will switch with the degree value). This is the rule of right triangles.

INQUIRY ACTIVITY SUMMARY

Basically what we did in the activity was use each special triangle and set their hypotenuses equal to 1 since the radius of a unit circle is 1. We did this by using the rules of special triangles and simplified. In the end we get the values for x and y and used those to find the ordered pairs that relate to the unit circle. The following walks through the steps for each individual special right triangle.

1). 30 Degrees Triangle:

• I first drew the triangle with the rules of special right triangles.

 

• I then simplified the three sides so that the hypotenuse was equal to one. In this case I divided everything by 2. We do this because the hypotenuse is really a radius in the unit circle and the radius of a unit circle is always 1.

• Next I labeled the hypotenuse 'r', the horizontal value 'x', and the vertical value 'y'. I also made note of their values.

 

• I drew a coordinate plane so that the triangle was in the first quadrant and the 30 degree angle was at the origin. I  then found the ordered pair of each vertex of the triangle using the horizontal and vertical values (x and y).

2). 45 Degree Angle: I did pretty much the same thing for the rest of the triangles as I did for the 30 degree triangle. The way of doing this is the same; it's just the number values that are different. For this triangle in particular we will start with the values 1, 1, and radical 2. Here are the pictures of my work:

Write out with rules of right trianglesSimplify to make hypotenuse equal to one

Label r, x, and y. Note values

Lastly, I put the triangle in the first quadrant with the degree at the origin. I then found the ordered pair of the vertices using the values of x and y.
 
 

60 Degree Triangle: This triangle in particular is very similar to the first triangle. The only difference with the 30 degree triangle is that x and y will be switched. The following shows how I did the activity with this triangle:

• Note how this triangle has the same results as the 30 degree triangle except the x and y values are switched. This is because a 30-60-90 and a 60-30-90 triangle are basically the same triangle just switched the other way.

How Does This help us derive the unit circle?

Well for starters, all three of the triangles help make up the unit circle; they are repeated in each quadrant in different directions. Also, the coordinates of the top vertices of each triangle are all points on the unit circle. These points will be repeated throughout the circle just as the triangles are repeated. So essentially, these triangles are what makes the unit circle what it is, since a circle is the set of all points equidistant to a point known as the center and, in a unit circle, the points are derived from these three triangles.

Quadrant Relevancy

The triangles we did in this activity were in the first quadrant. What happens to these triangles when we put them into quadrants II, III, and IV? The pictures below show the triangles in different quadrants:

From the top: 30 degree in Quadrant II, 45 degree in Quadrant III, and 60 degree in Quadrant IV 

 

The values switch positions and signs as the triangles switch quadrants. For example, the x in the 30 degree in quadrant II changed to the left side and therefore the x turned negative. The y went down in the 45 and 60 degrees in the III and IV quadrants making the y negative. Also in the 45 degree quadrant III the x turned negative too. All of these changes will make sense when filling out the unit circle.

INQUIRY ACTIVITY REFLECTION

1).Something I never noticed before about special right triangles is that they are used to derive the unit circle.

2).Being able to derive these patterns myself aids in my learning because I now have a foundation to the unit circle and I actually understand where it comes from so I'm not just memorizing something that I don't really know what it is.

 
 







RWA #1: Unit M Concept 5: Ellipses

1). An ellipse is the set of all points such that the sum of the distance from two points known as the foci is a constant. (http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/ellipse-drawn-from-definition-geogebra-dynamic-worksheet)

http://www.mathwarehouse.com/ellipse/equation-of-ellipse.php

2). Representing an Ellipse Graphically: An ellipse consists of a central point known as the center, 2 vertices connected with the major axis, and 2 co-vertices connected with the minor axis. The distance from the vertex to the center is known as 'a' and the distance from the co-vertex to the center is known as 'b'.

https://people.richland.edu/james/lecture/m116/conics/conics.html

Ellipses also have 2 points known as the foci. As demonstrated in the picture to the left, The sum of the distance from these two points (d1 + d2) is always a constant (2a). This is essentially what makes an ellipse. Also, the distance from the focus to the center is known as c..

http://coachjpocconics.weebly.com/ellipses.html

Algebraically:  As the picture suggests, there are two different ellipses with their corresponding equations which differ slightly with the placement of 'a' and 'b'. If 'a' (the bigger value) is under the y term, then the ellipse will be a vertical/ skinny ellipse and if it is under the x term, then the ellipse will be horizontal/ fat. 'H' and 'k' represent the x and y coordinates of the center.

Important: It is significant to know the eccentricity regarding an ellipse. Eccentricity is how round a conic is, 0 being perfectly round. An ellipse's eccentricity is greater than  0 but less than 1. This could be written as  e = 0 < x < 1. This is because an ellipse isn't perfectly round like a circle (whose eccentricity is 0) and yet it is more round than a parabola (whose eccentricity is 1). Eccentricity could be found by dividing 'c' by 'a' (c/a).

3). Real World Application: Elliptical Training Machinery
Elliptical gym equipment, such as elliptical runners/climbers are very useful when trying to avoid joint injury in a workout. Basically, the elliptical motion which is usually amplified with the use of handle bars or a motor drive, limits the impact on the joints that is very common with running. This article outlines this use for ellipses as well as a few other uses: http://www.ehow.com/info_8522010_real-life-uses-ellipses.html
...And this is a guy with a cool accent who will transmit elliptical knowledge to your brain in a brief 2 minutes:

4). URLs cited : Pictures from the top-down:
http://www.mathwarehouse.com/ellipse/equation-of-ellipse.php

https://people.richland.edu/james/lecture/m116/conics/conics.html

http://coachjpocconics.weebly.com/ellipses.html

Youtube video link: https://www.youtube.com/watch?v=AVrUEAiw5gA