I/D #1: Unit N Concept 7: How do SRTs and UCs relate?

Inquiry Activity Summary:

1. The 30*  triangle has sides labeled (r,x,y). "r" stands for the hypotenuse; "x" adjacent or horizontal value; and "y" opposite or vertical value. After labeling the triangle according to the rules of Special Right Triangles, (r equals 1 from 2x/2x. y equals 1/2 from x/2x. x equals rad3/2 from rad3*x/2x.) Take note that the three sides are simplified so that the hypotenuse equals 1 (the radius of the Unit Circle is always equal to 1). Then, the hypotenuse was labeled with "r", the adjacent side with "x" and the opposite with "y". Next, I drew a coordinate plane, origin being located at the labeled 30 degrees (0,0) (so the triangle could lie in quadrant 1). and all three vertices labeled as ordered pairs: 60*  (1/2, rad3/2) and the last vertice 90* as (rad3/2,0) .


 













2. The 45* triangle was labeled according to the rules of SRT, taking note that each value was divided by rad2 so that the hypotenuse equals 1 (the radius of the Unit Circle is always equal to 1) (longest side, r, stood for rad2/rad2 equal to 1, 1/rad2 equal to rad2/2 and 1/rad2 equal to rad2/2). I labeled the hypotenuse "r", horizontal value "x", and vertical value "y". Next, I drew the coordinate plane labeling x and y axis and the origin of (0,0)  at the labeled 45*, the next point at (rad2/2, rad2/2) and the last at (rad2/2, 0). 




 3. The 60* triangle was labeled according to the rules of SRT dividing each side by 2x so that the hypotenuse equals 1 (the radius of the Unit Circle is always equal to 1), ("r" equal to 1 from 2x/2x; "y" equal to rad3/2 from xrad3/2x; "x" equal to 1/2 from x/2x). Next, I labeled the hypotenuse "r", horizontal value "x" and the vertical value "y". Lastly, I drew the coordinate plane labeling the x and y axis, origin at (0,0) and the corresponding ordered pairs per vertice (at 30*: (1/2,rad3/2) and (1/2,0) at 90*. 

 












4. This activity helped me derive the Unit Circle because it allows you to visually take note of where the angles and ordered pairs from the UC come to be and how each are solved for.  For example, taking a closer look at the 45* triangle, we could note where its ordered pair came simplifying each side using the rules of SRT. 

 (http://jwilson.coe.uga.edu/EMAT6680Su12/Jackson/Writeup10DMJ/Unit%20Circle.PNG)

5.  The quadrant drawn in this activity lies in the first quadrant (note all four quadrants in Figure 2). If you draw the 30* triangle in quadrant 2 it's csc/sin are positive, sec/cos and tan/cot negative are negative. In the third quadrant, tan/cot are positive, and csc/sin, cos/sec are negative. Lastly, note that cos/sec are positive and csc/sin, tan/cot are negative.

 (http://www.google.com/imgres?client=firefox-beta&hs=hI5&sa=X&rls=org.mozilla%3Aen-US%3Aofficial&channel=sb&biw=1280&bih=673&tbm=isch&tbnid=1aKd0mo1Wb16-M%3A&imgrefurl=http%3A%2F%2Fwww.sparknotes.com%2Fmath%2Ftrigonometry%2Ftrigonometricfunctions%2Fsection3.rhtml&docid=3IGWql635sbzWM&imgurl=http%3A%2F%2Fimg.sparknotes.com%2Ffigures%2F0%2F067486b8a9659518b7099dac07405d29%2Fquadrantsigns.gif&w=210&h=210&ei=UlUJU5DHF7LlygGK8YGQDw&zoom=1&ved=0CFcQhBwwAQ&iact=rc&dur=442&page=1&start=0&ndsp=17) 

 

(https://encrypted-tbn0.gstatic.c/images?q=tbn:ANd9GcTokzujb26VtG3kOJjr5Hiq9rR1O7TWjkroQ8gv_peFYJc6oBNBUA)

Inquiry Reflection Activity: 

1. The coolest thing I learned from this activity was learning a basis for trigonometry using the rules for SRTs and UCs! Seeing how the curriculum for geometry and trigonometry tie together is fascinating!
2. This activity will help me in this unit because finding the value for the trig functions are more easily solved if you understand the unit circle in more depth. 
3. Something I never realized before about special right triangles and the unit circle is how they related with each other and the components that are used carefully to calculate each angle and degree that make up the unit circle.

References:

• http://jwilson.coe.uga.edu/EMAT6680Su12/Jackson/Writeup10DMJ/Unit%20Circle.PNG
• https://encrypted-tbn0.gstatic./images?q=tbn:ANd9GcTokzujb26VtG3kOJjr5Hiq9rR1O7TWjkroQ8gv_peFYJc6oBNBUA
• http://www.google.com/imgres?client=firefox-beta&hs=hI5&sa=X&rls=org.mozilla%3Aen-US%3Aofficial&channel=sb&biw=1280&bih=673&tbm=isch&tbnid=1aKd0mo1Wb16-M%3A&imgrefurl=http%3A%2F%2Fwww.sparknotes.com%2Fmath%2Ftrigonometry%2Ftrigonometricfunctions%2Fsection3.rhtml&docid=3IGWql635sbzWM&imgurl=http%3A%2F%2Fimg.sparknotes.com%2Ffigures%2F0%2F067486b8a9659518b7099dac07405d29%2Fquadrantsigns.gif&w=210&h=210&ei=UlUJU5DHF7LlygGK8YGQDw&zoom=1&ved=0CFcQhBwwAQ&iact=rc&dur=442&page=1&start=0&ndsp=17 
• http://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/8c85deff-b9ed-4c80-8b44-16e8e9322af2.png
• https://encrypted-tbn0.gstatic.c/images?q=tbn:ANd9GcTokzujb26VtG3kOJjr5Hiq9rR1O7TWjkroQ8gv_peFYJc6oBNBUA

RWA #1: Unit M Concept 4: Graphing parabolas given equation

1. Definition:     "The set of all points the same distance from a point and a line"

  ("http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/parabola-drawn-from-definition-geogebra-dynamic-worksheet")

 2. Properties:  

Algebraically:        

                       Equations/formulas-

                                                  Vertical graph:

Horizontal graph:

Graphically:

     A parabola is a graph that can go up, down, left or right. It has a center at it's vertex and focuses around the focus. The directrix is a line outside the parabola that is perpendicular to the axis of symmetry. The axis of symmetry is a line in the middle of the parabola where the focus, vertex and part of the directrix lie.

How to find it's key features algebraically and graphically:

     

     In order to put the equation into standard form you need to complete the square (note that only one term is squared). If the x term of the equation is squared and the value of p is negative it will go down and if the value of p is positive the graph will go up. If the y term is squared and the value of p is positive the graph will go right, if p is negative the graph will go left. The vertex is an ordered pair that is the center of the parabola. The values of h,k is the vertex. Remember that h always goes with x and y always goes with k.

Graphing a parabola!

("http://www.youtube.com/watch?v=AQngdAoPIgE")

     

Key features of a parabola!

 ("http://www.mathsisfun.com/geometry/parabola.html")

      The focus is an ordered pair that is a point inside the parabola that is algebraically found by subtracting the value p with the term of the vertex that is changing. The distance that the focus is from the vertex determines how skinny or how fat the parabola is. In addition, the distance from the focus to any point on the parabola to the directrix is always equal and that is called the eccentricity. A parabola's eccentricity is equal to 1 which is why the two distances are equal. The directrix is a line outside of the parabola that is x=# (vertical line) or y=# (horizontal line). The directrix is determined by subtracting  the value of p from the value that is not changing withing the vertex. "p" is a point that is determined by setting the term outside of the non-squared portion of the formula equal to 4p and solving. "p" is the value that determines how far away the focus and the directrix are from the vertex. The axis of symmetry is a line that lies in the middle of the parabola that is across the x or y axis and is x=#  or y=#. Notice that the value of the axis of symmetry is the x or y value between the vertex and focus that is not changing.  The focus, vertex, and a part of the directrix all lie on the axis of symmetry. 

3. Real World Application: Satellite Dishes!

("http://www.ips-intelligence.com/ips/wp-content/uploads/2013/08/2-satellite1.jpg")

 

("http://www.youtube.com/watch?v=fV9YuF__fM4")

     A satellite dish is an example of where parabolas are applied in the real world. Radio waves that are parallel to the axis of symmetry hit any curve on the surface and gets reflected off and directly to the focus.  The radio waves then create a signal when the wave concentrates off the focus. Note that there will not be a signal if the focus is not correctly built within the shape of the dish.

4. References

http://www.mathsisfun.com/geometry/parabola.html

http://www.sophia.org/tutorials/unit-m-concept-4a?cid=embedplaylist

http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/parabola-drawn-from-definition-geogebra-dynamic-worksheet

http://www.ips-intelligence.com/ips/wp-content/uploads/2013/08/2-satellite1.jpg

http://www.youtube.com/watch?v=fV9YuF__fM4

http://www.youtube.com/watch?v=AQngdAoPIgE

http://www.mathsisfun.com/geometry/parabola.html