Saturday, February 22, 2014

I/D# 1: Unit N Concept 7: How do SRT and UC relate?

INQUIRY ACTIVITY SUMMARY

1.        Describe the 30* triangle: The first thing I did was label according to the rules of the special right triangles. For the the 30* triangles, the hypotenuse is 2x, the opposite side is x, and the adjacent angle is xrad3. To simplify the 3 sides so that the hypotenuse is 1, 2x was divided by 2x to give me one. Since I divided the hypotenuse by 2x, I would have to divide the other two sides as well. The opposite side would be x divided by 2x, giving me 1/2. The adjacent side would be xrad3 divided by 2x to give me rad3/2. Next, I labeled the hypotenuse r, the horizontal value x and the vertical value y.

          I then drew a coordinate plane so that the origin is located at the labeled angle measure. The next step is to find the ordered pairs. The labeled angle is at the origin, so the ordered pair is (0,0). When looking at the 90* angle, we can see that the length of the x-side is rad3/2 and it lies on the x axis (the y -value is 0). This means the ordered pair would be (rad3/2, 0). Looking at the 60* angle, we notice that it the point is rad3/2 distance right of the origin and 1/2 distance above the origin, giving it the ordered pair (rad3/2, 1/2).

2.        Describe the 45* triangle: I labeled the triangle according to the rules of special right triangles. For the 45* triangle, the hypotenuse is xrad2, the horizontal side is x, and the vertical side is x as well. To make the hypotenuse 1, I divided xrad2 by xrad2 to give me one. Now I have to divide the two other sides by xrad2 as well. Since both are x, when I divide by xrad2, I get rad2/2. Then label the hypotenuse r, the horizontal value x and the vertical value y.
           Since the labeled angle, 45* is at the origin, the ordered pair is (0,0). The length of the x-side is rad2/2 and the point lies on the x-axis. This means the ordered pair would be (rad2/2, 0). Because the angle made from the hypotenuse and opposite side is rad2/2 right of the origin and the length or the vertical side is rad2/2,   the ordered pair would be (rad2/2, rad2/2).


3.        Describe the 60* triangle: This is very similar to the 30* triangle except the angles are flipped. The first thing I did was label according to the rules of the special right triangles. For the the 60* triangles, the hypotenuse is 2x, the opposite side is xrad3, and the adjacent angle is x. To simplify the 3 sides so that the hypotenuse is 1, 2x was divided by 2x to give me one. Since I divided the hypotenuse by 2x, I would have to divide the other two sides as well. The opposite side would be xrad3 divided by 2x, giving me rad3/2. The adjacent side would be x divided by 2x to give me 1/2. Next, I labeled the hypotenuse r, the horizontal value x and the vertical value y.

          I then drew a coordinate plane so that the origin is located at the labeled angle measure (60*). The next step is to find the ordered pairs. The labeled angle, 60*, is at the origin, so the ordered pair is (0,0). When looking at the 90* angle, we can see that the length of the x-side is 1/2 and it lies on the x axis (the y -value is 0). This means the ordered pair would be (1/2, 0). Looking at the 30* angle, we notice that it the point is 1/2 distance right of the origin and rad3/2 distance above the origin, giving it the ordered pair (1/2, rad3/2).

4.This activity helps us derive the UC by enabling us to understand the reasons to the values of certain points on the circle. The points on a unit circle is made from 30*, 45*, and 90* angles that are seen throughout all of the 4 quadrants. We now understand where the ordered pairs come from. Through simplifying the 3 sides of the triangle to make the hypotenuse equal to one, I was able to find the length of all three sides in proportion to each other. With this, I was able to find the 3 ordered pairs of the triangle. For all three triangles, the point formed from the joining of the hypotenuse and opposite angle represents a point on the unit circle. For the 30*, 45*, and 90* triangles, the ordered pairs of the previously explained point are (rad3/2, 1/2) (rad2/2, rad2,2) and (1/2, rad3/2). These three ordered pairs are repeatedly seen in all three quadrants. 
If you draw right triangles connecting the ordered pair to the origin on a unit circle, these triangles are identical to the triangles we encountered with this activity. Because of this activity, we now know where the ordered pairs come from, rather than just memorizing the graph, we are now able to explain why it is this certain way. We can also explain why the point at 0* is (1,0), the ordered pair at 90* is (0,1), at 180* is (-1,0), and the ordered pair at 270* is (0, -1) because the radius is 1.

 
5. The triangle in this activity lies in the first quadrant. The values of the the ordered pairs change as you draw it in the other 3 quadrants. For quadrant 2, all of the x values will be negative because it lies to the left of the origin. For quadrant 3, the x and y values will both become negative because the triangle lies to the left of the y axis and below the x axis. Only the y values will be negative in quadrant 4 because the triangles is below the x axis. 

This is a 30* triangle in Quadrant 2, note that the all x-values are negative.


This is a 45* triangle in Quadrant 3, note that both the x and y values are negative.
This is a 60* triangle in Quadrant 4, note that all the y-values are negative. 



INQUIRY ACTIVITY REFLECTION

1. The coolest thing I learned from this activity was how all the 4 quadrants follow the same patterns of special triangles, the only difference is that some are negative.
2. This activity will help me in this unit because I now understand where the values of the Unit Circle comes from, that the numbers are not random, and that you can find them using special right triangles.
3. Something I never realized before about special right triangles and the unit circle is how they are interconnected. I never noticed that you could be able to draw these special right triangles with in the Unit Circle and have the ordered pairs the same.

Friday, February 7, 2014

RWA# 1: Unit M Concept 5: Graphing Ellipses Given Equations and Defining All Parts

1."The set of all points such that the sum of the distance of two points, known as the foci, is a constant." (Kirch).
2.        The equation for ellipses is  . Graphically, an ellipse looks like a stretched out or elongated circle, it is oval shaped. Key features of ellipses include the center, major axis, minor axis, vertices, co-vertices, foci, and eccentricity. Graphically, the center is a point inside the ellipse which is the intersection of the major and minor axes, h,k. The next paragraph will go over the description of these key terms and on how to identify them graphically and mathematically.
           You can find these points algebraically by looking at the standard form of the ellipse. The x value will always be h and the y value will always be k, giving the center the point (h,k). "The major axis is the longest diameter of the ellipse."(http://www.mathopenref.com/ellipse.html) To determine if the major axis is horizontal or vertical , look at the standard form. It is horizontal if the bigger denominator is under the x^2 term and this means the ellipse will be fat. It will be vertical if the bigger denominator is under the y^2 term and the ellipse will be skinny. Algebraically, the bigger denominator is the a^2 term and the smaller, b^2. The major axis will be represented by a solid line and the length is 2a. Graphically, the minor axis is the shortest and is perpendicular to the major axis. This axis is represented by a dotted line. Algebraically, the length of the minor axis is 2b, meaning b distance from the center. Vertices are the points at the end of the major axis. To find this, plot a units to the left and right if x^2 has the bigger value under it. Plot it a units up or down if Y^2 has the bigger number under it. The vertices are expressed as 2 points that lie on the major axis.Co-vertices are the points at the end of the minor axis. You can plot the co-vertices b units up and down if x^2 has the bigger value under it or left and right if the Y^2 value has the bigger number under it. You can draw the ellipse by connecting all four points. The foci are the 2 points that define the ellipse. The closer the foci is to the center, the more circular the ellipse will be. The farther away the foci is from the center , the more it deviates from a circle, meaning its more stretched out. this gives it an eccentricity closer to 1. To plot the foci, plot c units to the left and right if x^2 has the bigger value under it. Plot it c units up and down if Y^2 has the bigger number under it. To find any missing values, use the formula: c^2=a^2-b^2.  Eccentricity is the measure of how much the conic section deviates from being circular. For ellipses, the eccentricity is between 0 and 1. You can find this by dividing c by a.











http://img.sparknotes.com/content/testprep/bookimgs/sat2/math2c/0006/ellipse.gif
The Picture shows the key points of ellipses, yay!

 
3.       Real world application of ellipses can be seen in the elliptical orbits in our solar system
People think that most objects in space that orbit something else move in circles, but that isn't the case. Although some objects follow circular orbits, most orbits are shaped more like stretched out circles/ovals. Mathematicians and astronomers refer to this oval shape as an ellipse. All of the planets in our solar system, satellites, and most moons have elliptical orbits.
         Earth moves around the sun in an elliptical orbit. Earth's orbit is almost a perfect circle!  its eccentricity is only 0.0167 which is very close to that of a circle, 0. Pluto has the least circular orbit of any of the planets in our Solar System (maybe because its not a planet! LOL). Pluto's orbit has an eccentricity of 0.2488. The Sun isn't quite at the center of a planet's elliptical orbit, instead, the sun is at the focus of the ellipse. "Because the Sun is at the focus, not the center, of the ellipse, the planet moves closer to and further away from the Sun every orbit." (http://www.windows2universe.org/physical_science/physics/mechanics/orbit/ellipse.html)
4. References:
http://cseligman.com/text/history/ellipses.htm
http://www.mathopenref.com/ellipse.html
http://www.windows2universe.org/physical_science/physics/mechanics/orbit/ellipse.html
http://www.youtube.com/watch?v=5nxT6LQhXLM