I/D #1: Unit N Concept 7 - How Do Special Right Triangles and the Unit Circle Relate?

Inquiry Activity Summary
1) The first and foremost part of interpreting a 30° special right triangle, is setting the side opposite of the 30° angle to x, and going along with that, the hypotenuse as 2x, and the other leg of the triangle as x√3. In order to align this set-up with the unit circle, we must set the hypotenuse equal to 1, since the radius of a unit circle will always be 1, and the hypotenuse will be exactly between the Unit Circle's center and its arc. So, if we set the hypotenuse to 1, then we basically divide all of our previously-acquired values by 2x, making it proportionate to the 2x = 1 ratio that we had fabricated. So, the x would become 1/2, and the x√3 would become √3/2.

2) In order to make sense of a 45° special right triangle, we should first set up the hypotenuse as x√2, and the other two sides as x (since both sides are the identical, according to geometry). To match this up with the unit circle, we set the hypotenuse to 1, which is most simply done, by dividing x√2 by x√2, which gives us 1. Dividing x by x√2 would make x times (1 / x√2), creating a situation in which the x's cancel out, and you are left with 1/√2. In order to rationalize that, you would multiply it by (√2 / √2), which would result in (√2 / 2) for your two other sides.

3) Lastly, the 60° special right triangle should include the exact same values as the 30° special right triangle would. All that one would have to do, is ensure that values accordingly match up with degrees. But, I suppose that I have enough time to explain this. Basically, the side opposite of your 60° angle would be set to x√3, your hypotenuse set to 2x, and your other leg, set to x. In order to substantiate a hypotenuse equal to 1, you would divide all of the values by 2x, and end up with 1 for the hypotenuse, √3/2 for the side that had previously equaled x√3, and 1/2 for the side that had previously been x.

4) How Does This Activity Help You To Derive The Unit Circle?
I suppose that this activity helped me more easily identify all of the values that encompass various common degrees in a unit circle, by listing out all of the possible values that I should know. If I were tasked to find out the coordinates of wherever any 30° or 60° angle's terminal side met with the Unit Circle's arc, there could only ever be three values that I need to know (their signs and or order depend upon which quadrant of the circle they are in): 1, √3/2, and 1/2. For the 45° angle, literally the only two values I would ever need to figure out coordinates of wherever the 45° angle's terminal side met with the unit circle's arc would be 1 and √2 / 2. This stuff is really useful in trigonometry and all.

5) The triangles drawn in this activity were in Quadrant I, ergo, their positive coordinates. The signs would change in differing quadrants, with Quadrant II implying a negative x-coordinate, Quadrant III, a negative x-coordinate and y-coordinate, and Quadrant IV, a negative y-coordinate.

Inquiry Activity Reflection
1) The coolest thing I learned from this activity was that from remembering three simple values (√3/2, 1/2, and √2 / 2), you can get everything that you need in the Unit Circle. All you'd have to do is remember what different quadrants do to the signs of the values, and all that other stuff.

2) This activity will help me in this unit because I can now, more easily remember the Unit Circle.

3) Something I never realized before special right triangles and the unit circle is that just by identifying which side is the longest of the two of a 30° or 60° angle, you can determine which has the √3/2, and which is the 1/2 (√3/2 is bigger than 1/2).

RWA #1: Unit M Concepts 4-6 - Conic Sections In Real Life

Parabola

1) Definition: The set of all points equidistant from a given point, known as the focus and a line, known as the directrix.

2) Algebraically: The algebraic equation that is used to standardize parabolas is as follows: (x-h)^2 = 4p(y-k) or (y-k)^2 = 4p(x-h). Basically, one would modify any given equation that is determined to be one of a parabola into this format, most usually, through completing the square. In order to distinguish equations of parabolas from equations from other conic sections, you basically need to know only one thing: one variable is square, while the other isn't. This is a distinct feature of the parabola that we don't find occurring within equations for circles, ellipses, or hyperbolas. Anyways, by looking at the value of p, and at which variable is squared, you may determine what the parabola would look like graphically, as in its orientation and all. 

Graphically: The first and foremost characteristic that one would find while examining the parabola, is that it looks awfully similar to the letter 'U'. Its orientation can be determined through the sign that is complementary with the value of p, and the variable that holds the square. If x is squared, then the parabola will either face up or down - up if the p is positive, and down if it is negative. If y is squared, then the parabola will either face right or left - right if the p is positive, and left if the p is negative. The point at which the parabola can be directly divided in half - that is, also considered its vertex - will be known as the vertex, and the line which divides it exactly in half, known as the line of symmetry. The focus point, which will be on the line of symmetry the distance of 'p' away from the center, will help one decide what the orientation of the graph is like - by way of being in the direction of the lines. Lastly, the focus point may also determine whether the parabola is fat or skinny - if it is close to the vertex, then it will be skinny; if it is far away from the vertex, it will be fat. AND the directrix is basically the opposite of the focus point, and is a line.

 
http://www.mathsisfun.com/geometry/images/parabola-formula.gif

Features Of: The vertex is most basically, (h,k), the focus, (h+-p,k) or (h,k+-p), depending on which variable has the square and what the value of p is, the line of symmetry, either h or k, again, depending on which value has the square, and p is the variable that precedes the variable without the square, divided by 4. Ah, and the directrix is the exact same thing as the focus point - save that it is determined by subtracting p if the focus is found through addition, and vice-versa - and, it is a line.

3) Real World Application: As demonstrated in the video below, there are many instances in real life in which parabolas occur. For example, when the teenager shoots the basketball into the basketball hoop, a parabola is seen - and even outlined, too. If it were practical, I suppose that basketball players would study up a bit more in math, in order to improve their shots.



4) Works Cited: http://mathworld.wolfram.com/Parabola.html
http://www.mathsisfun.com/geometry/images/parabola-formula.gif
https://www.youtube.com/watch?v=FXuTKpp-vME