ID #1: Unit N Concept 7: Knowing all Degrees and Radians Around the Unit Circle, Knowing all the Ordered Pairs Around the Unit Circle, Understanding and Applying ASTC to the Unit Circle

INQUIRY ACTIVITY SUMMARY

Many parts of this triangle have a specific label based on the Special Right Triangles rule. The first rule includes that the hypotenuse must equal 1 (ALWAYS). According to the 30, 60, 90 angles rules, the adjacent side to 30 (the side labeled x) has the value of x radical 3; the opposite side from the 30 degree angle (the side labeled y) has the value of 1/2; the hypotenuse of a 30 degree angle (the side labeled r) has the value of 2x . If this triangle were to be put in the unit circle across in the x- axis the origin (0,0) would be (radical 3/ 2, 0), and right above the previous point would be (radical 3/2, 1/2).

Along with the previous triangle, may rules apply to the sides and angles of the 45 degree triangle. The same rule about the hypotenuse applies to this triangle that it must equal 1 (labeled r) with the value of x radical 2. The adjacent side (labeled x) has the value of radical 2/2. the side opposite (labeled x) was valued the same of radical 2/2. The values are found when equaling the x's to the hypotenuse (x radical 2) leaving both of the sides equaling radical 2/2. If this triangle was inserted onto the unit circle, across from the origin (0,0) would lie the point (radical 2/2, 0), and above the previous point would be the point (radical 2/2, radical 2/2).

Just as the previous triangles, this 60 degree triangle includes rules to its sides and angles. The rule about the hypotenuse (labeled r) applies to this triangle with the value of 2x (the same as the 30 degree triangle). The side adjacent to the hypotenuse (labeled x) has the value of 1/2, and opposite from that side (labeled y) is valued radical 3/2. If this triangle were to be placed into the unit circle, across from the origin (0,0) on the x-axis would be point (1/2, 0), and right above would be point (1/2, radical 3/ 2).

                                                                                                               
All depending on where the ASTC is found on the circle, we will know which sin, cos, or cot is positive or negative. The ASTC starts from up right to the bottom from the left. Quadrant A (I) has ALL positive and NONE negative. Quadrant S (II), has csc/sin positive, and sec/cos, tan/cot negative. In quadrant T (III) tan/cot is positive, and csc/sin, cos/sec negative. Finally quadrant C (IV) has cos/sec positive and csc/sin, tan/cot negative. 

1. The coolest thing I learned from this activity was realizing that all along there were triangle involved that made the unit circle easier to understand. 
2. This activity will help me in this unit because it allowed me to find what i need to find using prior knowledge instead of just memorizing the entire unit circle. 
3. Something I never realized before about special right triangles and the unit circle is that they are connected through the point and radians values.

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

RWA Conic Section: Parabola

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

2.Properties:  

•                    Algebraically:  

•                            Formula/Equation: 

("www.mathwords.com")

•                        Graphically: The graph consist of many key factors that belong to a parabola. The center of the parabola is called the VERTEX and it focuses around the FOCUS. The AXIS OF SYMMETRY goes straight through the parabola, which is also perpendicular to the DIRECTRIX.  

•  The directrix is the line outside the parabola that you can identify by subtracting the value from the vertex (h or k) by p. The notation for the directrix is x=# and y=#. "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.To put the equation into standard form,  complete the square and be sure that only one term is squared. If the x term of the equation is squared and the value of p is negative the graph will go down and if the value of p is positive the graph will go up. In contrast,  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 as an ordered pair. Remember that h always goes with x and y always goes with k. 
• The axis of symmetry is a line that lies in the middle of the parabola that is written as x=#  or y=#.The value for the axis of symmetry is the number from the vertex that isn't changing.  The focus is an ordered pair found inside the parabola that is identified by adding the value of p with the term that isn't changing (h or k).
• 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.

4. RWA: Parabolic Antennas 

The  Parabolic Antennas are constructed as parabolas because that allows any sound and waves to hit it faster and easier than as a regular square would. he shape of he Parabola is used to attract waves. 

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