I/D #1: Unit N Concept 7: Knowing the degrees, radians, and ordered pairs of a Unit Circle

   The special right triangle (SRT) and the unit circle relate to each other in a way that the special right triangle of 30, 60, 90 and 45, 45, 90 can be found in the first quadrant of the unit circle. Moreover, it reveals the ordered pairs on the other three quadrants, depending on its sign (positive or negative). Special right triangles have a hypotenuse of 1 and the unit circle has a radius of 1. Thus, special right triangles can be located in all of the quadrants on the unit circle.

INQUIRY ACTIVITY SUMMARY

 1. 30 degree triangle

  This is the 30 degree special triangle. Across from the 30 degree angle is x, across from the 60 degree angle (not labeled) is x radical 3, and across from the 90 degree angle is 2x. The hypotenuse of this triangle is one.

  This pictures demonstrates a the sides and ordered pairs of the 30 degree special triangle. Since the the hypotenuse or the radius is equal to one, x (across from the 30 degree) has to be 1/2 and x radical 3 (across from 60 degree angle) has to be radical 3 over 2. The work is shown beneath the triangle. You can simply find the ordered pairs by looking at its points. This triangle is labeled at quadrant one on the graph. 

2. 45 degree triangle 

   This is a 45 degree special right triangle. Since two out of the three angles are 45 degrees, the letter x is the same. Across from the 45 degree is x and across from the 90 degree is x radical 2.

   This pictures reveals the ordered pairs and sides of the 45 degree special right triangle. Since the hypotenuse or radius is one, the letter x, across from the two 45 degrees, will be radical 2 over 2. The work is shown below the triangle. The triangle in this picture is located in quadrant one on the graph. Ordered pairs are plotted according to its location.

3. 60 degree triangle

   This is a 60 degree special right triangle. Across from 30 degrees (not labeled) is x, across from 60 degrees is x radical 3, and across from 90 degrees is 2x. This triangle is similar to the 30 degree special right triangle. However, you will noticed that the ordered pairs and sides will be switched.

   This picture shows the ordered pairs and sides of the 60 degree special triangle.  The hypotenuse/radius is one, which is similar to the last two special right triangles. The sides of this triangle is consisted of radical 2 over 2, 1/2, and 1. Work is shown below the triangle. Again, this triangle is labeled on the first quadrant. Ordered pairs are plotted according to its side and point.

4. This activity helps me derive the unit circle because it gives me the degrees, points, and ordered pairs that are located at quadrant one. Thus, with this knowledge, I can apply it to the other three quadrants on the graph. I can use the idea of reference angles to know what degrees  and ordered pairs (depending on the sign) will be on the unit circle for each quadrant. The special right triangle is part of the unit circle in a way that it depicts all the pieces that a unit circle has.

5. The triangles in this activity are located in quadrant one. Quadrant one is used to find all the other parts in the other quadrants. The "magic 5" relates to the other quadrants in terms of degrees, radians, and ordered pairs. The "magic 5" includes 0, 30, 45, 60, and 90 degrees. Knowing the first quadrant of the unit circle is very significant. Nevertheless, this is where the term reference angles and coterminal angles come in need. Coterminal angles are angles that have the same terminal side. Reference angles are positive, acute angles (from degree to x-axis).

http://01.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_34.gif

  This pictures shows the 30 degree special right triangle in quadrant II, III, and IV. All of these three quadrants are similar to the quadrant I. The difference in these quadrants is the ordered pairs because of the signs depending where it's located. Each quadrant has the same reference angle, which is 30 degrees.

 

http://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/ebaa19ac-ff8b-43a6-a793-a00d9ac15e86.png

   This picture shows the 40 degree right triangle in all of the quadrants on the unit circle. The only difference in each quadrant is the sides/ordered pairs because they can either be a positive or negative depending where it is plotted. Reference angles are still the same.

http://01.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_21.gif

   This pictures shows the 60 degree right triangle on the other quadrants (besides first quadrant). The sides are the same such as 1/2 still lies on the x-axis at quadrant III and IV. The difference, similar to the other special right triangles, is the sign on the ordered pairs. The values, otherwise, are still the same according to the first quadrant.

INQUIRY ACTIVITY REFLECTION

1. The coolest thing I learned from this activity was knowing how to relate special right triangles to unit circles.
2.  This activity will help me in this unit because I can now see the pattern and find a way to memorize all the degrees, radians, and ordered pairs in a unit circle.
3. Something I never realized before about special right triangles and the unit circle is how they are all connected to each other, in terms of the radius and the quadrants and all the other pieces that make up a unit circle.

References
http://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/ebaa19ac-ff8b-43a6-a793-a00d9ac15e86.png
http://01.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_34.gif
http://01.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_21.gif

RWA #1: Unit M Concept 5: Graphing Eclipses Given Equation

1. Mathematical Definition of an Eclipse: "the set of all points such that the sum of the distance from two points is a constant (Kirch)"

2.   The equation for an eclipse is . To define it graphically, an eclipse should look like an oval-shaped. There are two types of eclipses, which is either a "fat" or a "skinny" eclipse. In a "fat" eclipse,  the major axis has a length of 2a and the minor axis has a length of 2b. Moreover, the major axis lies according to the y-value since it has to stretch out horizontally. In a "skinny" eclipse, the major axis lies according to the x-value since it has to stretch out vertically. Both the "fat" and "skinny" eclipses' minor axis are opposite to its major axis. Graphically, an eclipse consists of the center (h,k), major axis, minor axis, foci points, vertices, and co-vertices. The center is the intersection of the major and minor axis. The vertices are endpoints to a major axis, while the co-vertices are endpoints to a minor axis. The foci are the focus points to demonstrate how much the eclipse deviates from being circular.
   To find the parts of an eclipse, you can determine by solving or putting the puzzle pieces together algebraically. In standard form, the x-value will go with h, and the y-value will go with k. The term a^2 and b^2 can either be under the term x and y depending what type of eclipse it is. The standard form of an eclipse will always be added and equaled to one. If it is not equal to one, make sure to simplify it. The standard form give you clues whether an eclipse is skinny or fat, center, major axis, minor axis, a, and b. Algebraically, the major axis will always be the bigger number and the bigger number will always be known as the term "a". Once you found your major axis, it is easy to find your minor axis by looking at the standard form. Thus, the term for minor axis is "b". To find the foci points or "c", you can use the equation a^2 - b^2 = c^2. Also, finding the eccentricity is very significant because it gives you an exact number of an eclipse that deviates from being circular. You can find the eccentricity by taking "c" and dividing it by "a" or c/a. The eccentricity of an eclipse is 0<e<1.


This image depicts the two types of ellipses and the key points both graphically and algebraically.

Here's a video about ellipses!!


3.  Ellipses can be found in our universe such as the solar system! The way the planets orbit around the sun has a shape of an ellipse including the satellites. Astronomers called this movement as elliptical orbits. For instance, the Earth's movement around the Sun has an eccentricity of 0.0167. In the solar system, the Sun is the focus. Since the Sun is the focus of an eclipse, the planets can either move closer or further away from the sun.
   Sometimes when you stare out of the sky, you can see the moon with a solar ellipse. This is "when the lunar disk passes directly between us and the sun." A lunar ellipse happens when the moon is considered a full moon. "Because the Moon casts a smaller shadow than Earth does, eclipses of the Sun tightly constrain where you can see them. If the Moon completely hides the Sun, even for a moment, the eclipse is considered total." The latest lunar ellipse was in December 2011 in Los Angeles. 

4. References:
http://www.teacherschoice.com.au/images/ellipse_types.gif
http://www.mathopenref.com/images/coordgeneralellipse/Equation-3.jpg
http://www.teacherschoice.com.au/images/ellipse_types.gif
http://www.windows2universe.org/physical_science/physics/mechanics/orbit/ellipse.html
http://www.skyandtelescope.com/observing/highlights/237963491.html