Saturday, February 22, 2014

I/D #1: Unit N: Concept 7: Unit Circle & Its Relation to Special Rights Triangle



We first had to label the triangle according to the rules of Special Right Triangle: the hypotenuse (r) would be 2x, the vertical side (y) would be x, and the horizontal side (x) would be x radical 3. Then in order to get the values of each side we have the simply the sides where the hypotenuse would equal 1. So we divide all the side by 2x. 2x/2x simplifies to 1, the hypotenuse. x/2x, the x's cancel out and leave 1/2. x radical three/2x, the x's cancel out, leaving radical 3/2. Next we drew the coordinate plane with the triangle lying in quadrant 1. Now the origin will be (0,0). The coordinates for 90* angle will be (radical 3/2, 0) since the distance between the origin and the 90* is radical 3/2. The outermost angle (60*) will be (radical 3/2, 1/2) because now we move up y which equals 1/2. 

We follow the same steps we did before but this time it's 45* which means two side will be the same since the other angle, besides 90*, will also be 45*. The hypotenuse (r) would be x radical 2, the vertical side (y) would equal x as well the horizontal side (x). Again we want to simplify the sides so the hypotenuse would equal 1. Now in order to do this, we divide the hypotenuse by itself to get 1, and we divide x by x radical 2. But by doing this, we leave a radical in the denominator and this is a major no no. So we rationalize by multiply radical 2 on both sides and it simplifies to radical 2/2. Again the origin will be (0,0). The 90* angle will be (radical 2/2, 0) since x's distance is radical 2/2. And the remaining angle will be (radical 2/2, radical 2/2 since y's distance is also radical 2/2. 

Last angle! The hypotenuse (r) is 2x - like the 30* triangle. The vertical side (y) is x radical 3 and the horizontal side (x) is x. Now again we must simplify the sides in order to have the hypotenuse equal 1 so we divide all sides by 2x. (y)'s x radical 3/2x will simplify to radical 2/2 while (x)'s x will simplify to be 1/2, the opposite of 30*. Once again the origin will be (0,0). 90* angle will be (1/2, 0) since x's distance is 1/2 and the last angle will be (1/2, radical 3/2) since y's distance from x is radical 3/2.

This activity was able to help me derive the Unit Circle by providing the angles for the first quadrant of the circle and establishing a foundation for the Unit Circle. Note that the coordinates for the 30*, 45*, 60*, in the first quadrant are the same as the ones I had mentioned previously, and they are also all positive! 

As I had mentioned, the triangles lied in the first quadrant, where both x and y are positive (+) The values change if drawn in quadrant 2, 3, or 4 by sign changes meaning the value being positive or negative. 

This is the 60* triangle drawn in the 2nd quadrant. Everything is still the same as it was in the 1st quadrant, only now the x value is negative due to the quadrant's location and the knowledge of positive and negative components on a graph. Because x is negative so is the x-coordinate in the 2nd quadrant.

x: negative (-) | y: positive (+)

This is the 45* triangle drawn in the third quadrant. It is still the same as the 45* triangle drawn in the 1st quadrant only flipped and now both x and y are negative (again this is due to basic knowledge of the positive and negative components of a graph) 

BOTH (-)

x: negative | y: negative

And lastly, this is the 30* triangle. Once again, it is made of the same components as the triangle in the 1st quadrant only the value of positive and negative changed. This time y is negative due to being located in quadrant 3, meaning that the y coordinate will be -1/2. X will remain positive. 

x: positive (+) | y: negative (-)


The coolest thing I learned from this activity was that you really only need to know the first three angles (knowing the quadrant angles would also be helpful) but if you have learned the magic three, it will be incorporated into the other quadrants and it's only a matter of logically knowing the positive and negative values of each quadrant. 

 This activity will help me in this unit because from here I can derive the unit circle if my mind draws a sudden blank. The outermost coordinates of the triangle gives us the coordinates to the angles in the circle and these three triangles are repeated four times throughout the circle and only certain coordinate values will change depending on their placement on the graph, more specifically it depends on the quadrant the angles are located in. Knowing this and the special right triangles coordinates will significantly help with the unit circle and the future concepts such as in finding the exact values of all six trig. functions. 

Something I never realized about special right triangles and the unit circle is that the special right triangles when layered over each other with the all three origins at (0,0) create a circle, it blew my mind that I couldn't make this connection but now that I see the special right triangles is intertwined with unit circles in creating the circle.

Monday, February 10, 2014

RWA #1: Unit M Concepts 4-6: Conic Sections in real life.

Ellipses (Concept 5)

1. The mathematical definition of an ellipse, according to Mrs. Kirch is, "the set of all points such that the sum of the distance from two points in a constant."

2. Key features of an ellipse include the center, major and minor axis, two vertexes, two foci, two co-vertices, and the ellipse's eccentricity. 

This conic section algebraically looks like:  

See full image here 

The ellipse is cut by a slant through a cone, it cuts a shape of an elongated circle, an oval. Here are some photos and video for you reference. 

See first image here (graph)
See second image here (cone)


Graphically you can easily see if your ellipse will be skinny or fat visually. If it is skinny, you will know your major axis will be x = and if it is fat, you will know your major axis will be y =. You can determine your center by finding the point at which the major and minor axis intersect and it will be defined as (h,k). H always goes with x and k always goes with y. The vertices, co-vertices, and foci will be easy to identify on a graph. Remember that the vertices, foci, and major axis will all have a certain number in common.


You can find multiple key features from the standard form of the equation alone. The center will always be (h,k) and h will always be with x and k will always be with y. 


 The bigger denominator will be a^2 and the smaller denomiator will be b^2. The bigger denominator will determine if your ellipse will be horizontal "fat" or vertical "skinny." You only need to see if the bigger denominator is under the x^2 term, then it will be horizontal. If it is under the y^2 term it is going to be vertical. A fun little trick to remember if it will be vertical is "(wh)Y you so skinny?The major axis is the factor in making the ellipse horizontal or fat since it is the longest diameter, the minor axis is the shortest by default. If the ellipse is horizontal the major axis will be y = and if it is vertical the major axis will be x = [another neat trick, "(wh)Y you so X-tra skinny?" since if it is vertical and skinny, the major axis will be x.] 


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. The bigger denominator's (a) square root will determine how many units away from the center you will plot the vertices and the distance between the two vertices (the major axis, usually drawn with a solid line graphically) will equal 2a. Plot a units to the left and right of the center if x^2  has the bigger value under it. Plot  a units up or down from the center if y^2 has the bigger number under it. The smaller denominator's (b) square root will also determine how far away you will plot the co-vertices from the center and the distance between the two co-vertices (the minor axis, which is perpendicular to the major axis graphically. Also typically drawn dotted) will equal 2b. Plot the co-vertices b units up and down from the center if x^2 has the bigger value under it or left and right from the center if the y^2 value has the bigger number under it.  The foci can be found with the equation: a^2 - b^2 = c^2. C (foci) is also associated with the major axis and vertices, and will have some of the same coordinates in common. 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. Now that we have "c" we can find the ellipse's eccentricity with the equation "c/a" and an ellipses's eccentricity is 0 < e < 1. The closer an the eccentricity is to 1, the skinnier the ellipse will be, the farther it is from 1 (closer to 0) the fatter the ellipse will be.  

For more clarification please reference to this video giving you a quick introduction to ellipses. 

3. A real world application of an ellipse can be seen in architecture through "whispering galleries." One of the properties that an ellipse has is a reflective one. "If a tangent line were drawn at point P, the angles formed by that line and the lines PF and PF would be equal. Thus a light or sound wave emanating from one focus of an elliptic surface would be reflected to the other focus." ( //associated with the last image. A most famous example of such is in St. Paul's Cathedral in London. 

 Based on these two images, we can see that at the foci of the ellipse, sound is reflected from one focus to another through a direct line (the major axis.) But an essential key to all of this is the shape of course. The ellipse's curves are able to reflect soundwaves that hit it and bounces it off the walls, focusing them to several locations in the room. However if you were to stand at the very endpoints of the ellipse, the range of the soundwaves would not be as far or wide and would not reflect as well as the foci. And should you stand at the co-vertices, the results would be less desirable since the co-vertices fall on the minor axis and would not provide a sufficient range for the soundwaves to reflect off of. So if you are an eavesdropper and you're standing in the right place, you can hear a slightly audible conversation happening maybe 30-40 feet away.

4. Works Cited