Friday, February 21, 2014

I/D#1: Unit N Concept 7-9: Knowing all degrees,radians,and ordered pairs around the unit circle;understanding and applying ASTC to the unit circle;finding exact values of all 6 trig functions when given angle in degrees or radians;finding angles when given exact value of any 6 trig functions

INQUIRY ACTIVITY SUMMARY
Numbers #1-3 are preformed in the video.

4.This activity helps you in deriving the unit circle because the "y's" vertice(the top vertice) ends up being the ordered pair for  the 30,45, 60 degree in the unit circle and those are you main repeated ordered pair of the unit circle, except the other ordered pairs are mirrored. 
5. The triangle drawn in this activity are located in quadrant 1. For quadrant 2, quadrant 1 is mirrored to the left, but all of the x values are negative. For quadrant 3, quadrant 2 is mirrored down, but both the x and y values are negative. For quadrant 4, quadrant 1 is mirrored down, but the y values are negative.
In quadrant 2, the 30 degree triangle just like folded or mirrored to the left, which made the x value become negative because looking at it as in a coordinate plane, the left is in the negative. In quadrant 3 the 45 degree triangle mirrored to the left, but then mirrored down. Therefore, both the x and y values are negative because it is down and left. In quadrant 3, the 60 degree just mirrored down or if you want you can see it as if it mirrored to the left, then mirrored down, and finally mirrored to the right to make a complete circle. The x value remained positive, but the y is in the bottom right, which is negative.
INQUIRY ACTIVITY REFLECTION
1. The coolest thing I learned from this activity was the unit circle is formed by 30,45,and 60 right triangles and the unit circle's ordered pairs are all the vertices of those triangles.
2.This activity will help me in this unit  because first I will not be mandated to remember all of the ordered pairs and if  I decided to just remember the ordered pairs and I forget the ordered pairs of the unit I could just see the triangles and find their vertices and those will be my ordered pairs.
 3.Something I never realized before about special right triangles and the unit circle is that the triangles vertices make up the unit circle and before I never applied special right triangles to anything I would just do the soh cah toa with special right triangles and now the special right triangles are applied to the unit circle.

Thursday, February 6, 2014

RWA#1: Unit M Concept 5: Graphing ellipses given equation and identifying all parts(center,focus,major and minor axis, vertices and covertices,and eccentricity)

1. "The set of all points such that the sum of the distance of two points is a constant."
2. An ellipse in standard form will have the equation that looks like (x-h)^2/a^2+(y-k)^2/b^2=1 or (x-h)^2/b^2+(y-k)^2/a^2. If the bigger number is under the x, then the ellipse is going to be fat. If the bigger number is under the y, then is going to be skinny. One way to remember is "y so skinny" and "you are x-tra large." An ellipse looks like a squished out/up circle. An ellipse has all of the following:center,focus,major and minor axis, vertices and covertices,and eccentricity. If you are given the general equation of an ellipse you are going to have to complete the square(it has to equal one) in order to get the equation into standard form. Once, you have it into standard form you can right away get the center. Remember that x goes if h and y goes with k and the signs will switch. Now looking at the denominator of the equation you have to see if the bigger number is under the x, then the ellipse is going to be fat and if the bigger number is under the y, then it is going to be skinny. Again looking back at the denominator of the equation, the bigger number is always "a" but you need to take the square root of that number. The other number is "b" and you also take the square root of that number. In order to find "c" you will need to plug in the two variables that you know into the formula a^2-b^2=c^2 and solve for c. To find the vertices you will have to look at your center and whether your graph is going to be fat or skinny. If your graph is fat then then you add and subtract "a"to x of the center and y will stay the same and vice versa goes if your graph will be skinny. Therefore, your major axis will be y= the y of the center(the major axis has the vertices,center, and focus)and vice versa goes if your graph will be skinny. To find the co vertices you will have to look at your center and whether your graph is going to be fat or skinny. If your graph is fat then then you add and subtract "b" to y of the center and x will stay the same and vice versa goes if your graph will be skinny. Therefore, your minor axis will be x= the x of the center and vice versa goes if your graph will be skinny. Major axis are solid lines and minor axis are dotted lines. To find the eccentricity you plug in "c" and "a" into the formula:e=c/a. In order to find the foci you have to look at your center and whether your graph is going to be fat or skinny. If your graph is fat then then you add and subtract "c"to the x of the center and y will stay the same and vice versa goes if your graph will be skinny.An ellipse has to have an eccentricity of greater than zero but less than one. Eccentricity is just "a measure of how much the conic section deviates from being circular."  "As the distance between foci increases the eccentricity increases, or the reverse relationship."
To get more information on how to find the parts of an ellipse watch below.







3. A real world application of an ellipse is an extracorporeal shockwave lithotripsy, which "enables doctors to treat kidney and gall stones without open surgery"(http://www.lupdirect.com/urologicalprocedures_lithotripsy.php)
Basically,external, high-intensity, and focused shock waves  pass through the body and they reach the  stones and the pressure causes the stone to be stressed. Eventually, they  fracture into smaller fragments that can be pass by the patient easy. Ultrasound or fluoroscopic x-ray system is used to direct the focus of the waves accurately on the stone. This is better than a doctor performing open surgery, which doctors can damage surrounding tissue and the risk of infections.
The extracorporeal shockwave lithotripsy has half of a  three dimensional  representation of an ellipse piece that is sitting on the patient's side. The lithotripter works because of the reflectiveness of an ellipse.
4.Works Cited
  • http://www.lessonpaths.com/learn/i/unit-m-conic-sections-in-real-life/conic-sections-in-real-life
  • http://www.lupdirect.com/urologicalprocedures_lithotripsy.php
  • http://www.sophia.org/tutorials/unit-m-concept-5a?cid=embedplaylist