Friday, February 28, 2014

I/D# 1: Unit N Concepts 7-9 : How Do Special Right Triangles and the Unit Circle Relate?

INQUIRY ACTIVITY SUMMARY
1. Describe a 30 degree Angle 
      I first labeled the triangle according to the Special Right Triangle Rules which is the hypotenuse 2x, the horizontal value x radical 3, and the vertical value x. I then had to simplify the three sides of the triangle fairly in a way such that the hypotenuse=1 so i divided each side by 2x (which is the hypotenuse) giving me: r=1, x=radical 3 over 2, and y= 1/2. I labeled the hypotenuse r, the horizontal value x, and the vertical value y. In order to draw a coordinate plane for each angle i first put the ordered pair (0,0) at the 30 degree corner, then since our x= radical 3 over 2, i knew my ordered pair for the 90 degree angle was (radical 3 over 2,0) because it didn't move up the y axis. For the 60 degree angle, i simply already knew that x=radical 3 over 2 and our y=1/2 so i put them as the ordered pair (radical 3 over 2, 1/2) since it moved x units to the right and y units up.

2. Describe a 45 Degree Angle 
      I first labeled the triangle according to Special Right Triangle Rules which is the hypotenuse x radical 2, the horizontal value x, and the vertical value x as well. In order to have the hypotenuse equal to one as  i divided each side by the value of the hypotenuse which is x radical 2 resulting in as r=1, x=radical 2 over 2, y=radical 2 over 2. I labeled the hypotenuse r, the horizontal value x, and the vertical value y. To find the coordinate planes, i applied the same concept as i did for the 30 degree angle above. I put the ordered pair (0,0) at the 45 degree left corner, then since our x= radical 2 over 2, i knew my ordered pair for the 90 degree angle was (radical 2 over 2 over 2,0) because we didn't move up the y axis. For the other 45 degree angle, i simply already knew that x=radical 2 over 2 and our y=radical 2 over 2, so i put them as the ordered pair (radical 2 over 2, radical 2 over 2) because we moved x units to the right and y units up.


3. Describe a 60 degree Angle
      I first labeled the triangle according to Special Right Triangle Rules which is the hypotenuse 2x, the horizontal value x, and the vertical value x radical 3. To have the hypotenuse equaling to one, i divided it by itself and the other sides as well having r=1, x=1/2, y=radical 3 over 2.  I labeled the hypotenuse r, the horizontal value x, and the vertical value y.To find the coordinate planes, i applied the same concept as i did for the above angles. I put the ordered pair (0,0) at the 60 degree left corner, then since our x= 1/2 , i knew my ordered pair for the 90 degree angle was (radical 2 over 2 over 2,0) because we didn't move up the y axis and moved x units to the right. For the 30 degree angle, i simply already knew that x=1/2 and our y= radical 3 over 2, so i put them as the ordered pair (radical 2 over 2, radical 2 over 2) because we moved x units to the right and y units up.


4. This activity helped me derive the unit circle because it gave me the coordinates, points, and angles that are within the unit circle as seen below.  I understood the foundation of it, allowing me to not only have to memorize numbers but know why those numbers exist. I thought it was important to do this activity because it gave me the basic comprehension of what a unit circle is.

5. The 30, 45, and 60 degree angles are within the first quadrant of the unit circle. The second quadrant is a reflection of the first, the third is a reflection of the second, and the fourth is a reflection of the third. In the first quadrant, all coordinates are positive, in the second quadrant, only the x value is negative, in the third quadrant both x and y are negative and in the fourth quadrant only the y is negative having that the first and third quadrant are positive and the second and fourth are negative. For a better understanding look below.








Inquiry Reflection Activity
1. The coolest thing I learned from this activity was how each quadrant is reflected with triangles.
2. This activity will help me in this unit because understanding and memozing the unit circle is a basic need to solve problems in all the concepts.
3. Something I never realized before about special right triangles and the unit circle is that the whole unit circle consists of many right right triangles that give it its coordinates.



Tuesday, February 11, 2014

RWA#1:Unit M Concepts 4-6: Graphing and Identifying all Parts of an Ellipse


1) Definition
"The set of all points such that the sum of the distance from two                   points is a constant." (Crystal Kirch) 

2)


http://www.mesacc.edu/~marfv02121/readings/conics/conics-2.GIF


      The above equations show an ellipse in standard form also meaning both term terms are squared, added, and have different coefficients. In standard from we can find the vertex, by using the h and k, and we can find our a and b by taking the square root of the a and b in the standard form. a is the distance in units away from the center and the vertex and b is the distance in units from the center to the co vertices. you can also use b to find the co-vertices which are the endpoints of the minor axis. To find c you must use the equation a^2 -b^2=c^2. c is the distance in units from the center to the foci. the c will also be a part of the foci. The below pictures give a visual. 
     
Horizontal 
http://theo.x10hosting.com/examples/Ellipse/Ellipse6.jpg

Vertical 
http://theo.x10hosting.com/examples/Ellipse/Ellipse7.jpg
     The following parts to graph an ellipse are the:  foci's, major axis, minor axis, and the eccentricity. The foci are spaced equally on each side of the center and they always lies on the major axis.The closer the foci is from the end point the skinnier the graph gets. To graph the foci you must plot the amount of c units to the right/left if x is squared but if y is squared then plot them up/down. identify whether the major axis is horizontal or vertical you must see if the bigger denominator in under the "y" squared or the "x" squared. If it is the bigger denominator is under the "x" squared term than it is horizontal and if its under the "y" squared term than it is vertical. The eccentricity of an ellipse (the measurement of how much a conic section varies from being a circle and for must be) must be greater than 0 but less than one. To find it you divide c and a. For more detailed instructions see:  http://jwilson.coe.uga.edu/EMAT6680/Brown/6690/InstrUnit/DayFive.htm

For better instructions on how to graph an ellipse watch the video below.





3) Real World Application 
Our solar system is an example of an ellipse. You may think that most objects in space orbit in circles, however that is not the case. Although some do follow circular orbits most orbits are shaped like a stretched out circle. All of the planets in our solar system move along elliptical orbits having the sun as a foci. The orbits of the moon and satellites of the moon are also elliptical. (http://www.windows2universe.org/physical_science/physics/mechanics/orbit/ellipse.html)

4) Works cited