Saturday, February 22, 2014

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

Inquiry Activity Summary


  1. The 30­° Triangle 
(http://www.gradeamathhelp.com/image-files/30-60-90-triangle.jpg)








         
     
                Before we begin to solve the special right triangle section of the unit circle, we must first follow the special right triangle rules as shown in the first picture. For this triangle, we start from the 30° angle and move from there. So the hypotenuse will stay the same as the 2 value, but the side opposite from our angle will be the 1 value because that is what is next to the x in our rule and the side that will be horizontal will have a set radical 3 value. We will also pretend the triangle is on a coordinate grid where the horizontal side is basically the y value and the vertical side is the x while the hypotenuse will take the radius value.
              Now we can solve for this special right triangle with the parameters given. Since we are solving for a section of a unit circle, the radius always has to be one so there is one value given. To get to the value, we must divide the hypotenuse by 2 and do the same for each side. The y value becomes 1/2 and the x value becomes radical 3 over 2. The coordinates are easy to plot if you are still imagining a coordinate grid and are shown next to the corners of the triangle, with our origin starting at the 30°. 


   2. The 45° Triangle

(http://www.gradeamathhelp.com/image-files/45-45-90-triangle.jpg)







           

                 What we did previously will now be used to solve for this triangle. Now we have a 45 45 90 right triangle and we are starting from one of the 45° angles. Use the new set of special right triangle rule for this triangle and now we have to divide the hypotenuse by radical 2 to get our desired one value. Do that division for each side and now You have your x and y values as labelled in the right picture. Remember you cannot have a radical in the denominator so make sure you multiply by a radical over radical value to get rid of that radical. This time the ordered pairs are circled a midst all the work that is shown. 




  3. The 60° Triangle


(http://www.gradeamathhelp.com/image-files/30-60-90-triangle.jpg)


              Truth be told the resourceful learner would just use the values solved for in the 30° triangle and switch them accordingly. But an explanation will be provided for the slower learners in the crowd. We use the same rule for our 60° triangle as the first one because it is apart of the same 30,60,90 triangle, and we still divide the hypotenuse by 2 to get our desired 1 value and do the same thing for the other sides. But now we must switch the x and y values we had because the angle we are starting from is different. We must also pay careful attention to our coordinates because our values have been switched. The coordinates are circled again and the x and y values have been changed so make sure you read carefully.

 4. How does this help you derive the unit circle

        This triangle activity helped me figure out why the coordinates change as I go through a unit circle. It makes perfect sense now that I look at it, the unit circle is just composed of a bunch of triangles. The coordinate we care about is the one that is connected to the y value and the hypotenuse, that is where I figured out why we have such crazy numbers like radical 3 over 2. It also helps me remember what coordinates go where for the unit circle because now I look at the unit circle like a grid where the radius is always 1. Here is a picture I found that better illustrates the idea that these triangles fit in the unit circle. 
(http://upload.wikimedia.org/wikipedia/commons/thumb/9/94/Special_right_triangles_for_trig.svg/220px-Special_right_triangles_for_trig.svg.png)

 5. What quadrant does the triangle drawn in this activity lie in?  How do the values change if you draw the triangles in Quadrant II, III, or IV?  Re-draw the three triangles, but this time put one of the triangles in Quadrant II, one in Quadrant III, and one in Quadrant IV.  Label them as you did in the activity and describe the changes that occur.

The triangles drawn in this activity belong in quadrant I because we start from the origin and go right on the x axis and our triangle points upwards into quadrant I. The values change signs whenever we move them into other quadrants, but as long as we keep the same angle, the values will stay the same. For example the x and y value will be negative in quadrant III and only the y value will be negative in quadrant IV. Lets draw a each angle in a different spot to see these differences.

Here we have a 45° angle in quadrant II. The only thing that changes here is the sign for the x value and you can clearly see that the coordinate changes accordingly. The x and y value are labelled here to show you the difference. 

Here we have a 30° angle in quadrant III. This time both the x and the y value change signs as represented in the image, The coordinate value changes signs while in this quadrant as well. 


Here we have a 60° angle in quadrant IV and this time only the y value changes signs as seen in the labels above.

Inquiry Activity Reflection

The coolest thing I learned from this activity was how the unit circle changes values according to rules for a triangle. I have always wondered about the similarity between triangles and circle when eating a pizza but I've never really put too much thought into it.

This activity will help me in this unit because simply put, it will make filling in the unit circle much easier. Although I still like memorizing it a lot more than this.

Something I never realized before about special right triangles and the unit circle is how closely related they are to each other. Special right triangles are the reason we have such wonky numbers in our unit circle when we are filling out the coordinates. 


Monday, February 10, 2014

RWA1: Unit M Concepts 4-6: Conic Sections in Real Life


RWA: Ellipse Conic Section

  1. Definition: "The set of all points such that the sum of the distance from two points is a constant." (http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/ellipse-drawn-from-definition-geogebra-dynamic-worksheet)
  2. Description: 
  • Algebraically: 
    (http://www.clausentech.com/lchs/dclausen/algebra2/ellipses.htm)
    Two different ways the equation can be written, both producing an ellipse. How each is read and graphed will be explained in more detail later.







  • Graphically: Ellipse is an oval-shaped graph that has a major solid axis and a dotted minor axis. There is a focus graphed inside the ellipse and the center lies, obviously, in the center. Refer to the picture to figure out whether the ellipse will be horizontal or vertical. Points are also graphed on the endpoints of both axis. 
  • Key Features: 



  •  http://media.wiley.com/Lux/94/219994.image1.jpgHere are all the key features of an ellipse. The center is not listed here but it is obviously the center and is found by the simple h.k rule from the equation, just take the h,k value and the opposite signs of those numbers are your center. The way you find the vertex is taking the square root of the bigger value or the "a" value and adding it and subtracting it either the h or k value whether its horizontal or vertical, view the previous image to better understand. The minor axis is found the same way but with the smaller "b" value. Both of these are found as the denominators and have axis that cross through them all the way to the center. The axis that crosses the vertex is usually solid and is called the major axis. The one that crosses the co-vertex is called the minor axis and is usually dotted. For more information visit http://www.clausentech.com/lchs/dclausen/algebra2/ellipses.htm .
  • Foci and Eccentricity: The foci are found by finding the c value through the equation a^2-b^2=c^2 where a must be greater than b. The value of c is then added and subtracted to either the h or k value in the same way the a value was added and subtracted. Foci dictate how circular the ellipse is, if it is closer to the center then it is more circular. The eccentricity or the value of c/a will always be less than one for an ellipse. 
  • Further Help on Graphing an Ellipse: 


3. Real World Application: Probably the most common occurence of ellipses is in space. The way gravity works makes orbit travel not in a perfect circle, but in a fixed ellipse. 

(http://britton.disted.camosun.bc.ca/elliorbit.GIF)
One perfect example of this fixed path is Halley's Comet, which was discovered by Edmund Halley. "Halley's Comet takes about 76 years to travel abound our sun. Edmund Halley saw the comet in 1682 and correctly predicted its return in 1759. Although he did not live long enough to see his prediction come true, the comet is named in his honour."(http://britton.disted.camosun.bc.ca/jbconics.htm) Halley's Comet chooses this path in a way that it cuts through space like a conic section would normally do. Here is a video describing the comet in full detail and a better view of it's ellipse path.



4. Works Cited: