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Friday, February 21, 2014

I/D #1 - Unit N Concept 7: How do SRTs & the UC relate?

Inquiry Activity Summary:
The unit circle activity is to help us develop our understanding of the special right triangles and how it correlates with the unit circle. There are three special right triangles: 30º, 45º, and 90º. The three sides of the triangle has to be simplified in such a way that the hypotenuse is equal to one because when inserting the triangle into the unit circle, the hypotenuse would be the radius, which always have to be one.

1. Rules for 30º triangle:
  • Hypotenuse (r) is 2x *Remember to change it to 1 by dividing 2x*  
  • Horizontal value (x) is x *When divided by 2x, it will make 1/2* 
  • Vertical value (y) is x radical 3 *When divided by 2x, it will be radical 3/2*
  • The 30º angle must be at the origin (0,0)

2. Rules for 45º triangle:
  • Hypotenuse is radical 2 but remember to divide by radical 2 to make it 1
  • Horizontal value is 1 over radical 2, when divided it'll be radical 2 /2
  • Vertical value will be the same as horizontal value
  • 45º angle must be at origin

3. Rules for 60º triangle:
  • Hypotenuse is 2x, divide by 2x and it'll become 1
  • Horizontal value is x radical 3, after dividing it'll be radical 3 / 2
  • Vertical value is x, after dividing it'll be 1/2
  • 60º must be at origin

4. The special right triangles activity helps us derive the unit circle because the points from the triangles will form a circle if all the points are connected. But of course, the 30º, 45º, & 60º are only in the first quadrant. So, you would have to find the other coordinate pairs, however, once you know quadrant 1, you basically know all the other quadrants.

5. The triangles drawn in this activity lies in quadrant 1. The values will change with positive or negative depending on which quadrant you're in. If you're in the 2nd quadrant, then your x value will be negative and your y value is positive. If it's in the 3rd, both the x and y value will be negative. If it's in the 4th, x value will be positive and y value negative.
In this 2nd quadrant, I showed the 30º reference angle,
the actual angle is 150º. As you can see, the coordinate pair
farthest to the right, is similar to the 30º angle in quadrant I.
The only difference is the x value is negative.
In this 3rd quadrant, I showed the 45º reference angle, the angle
is 225º. The coordinate pair closest to the bottom has both a negative
y and x value. It's also similar to 45º in quadrant I.
In quadrant IV, I showed the 60º reference angle and the angle
would be 300º. The coordinate pair closest to the bottom has a negative
y value and a positive x value. This angle is similar to the 60º angle
in quadrant I.

Inquiry Activity Reflection:
"The coolest thing I learned from this activity was..." The unit circle consists of special right triangles and it's not just a bunch of numbers you have to memorize; there's a pattern.

"This activity will help me in this unit because..." I'll be able to comprehend the unit circle without having to memorize random points.

"Something I never realized before about special right triangles and the unit circle is..." That just by knowing the first quadrant, you pretty much memorized the entire unit circle.

Monday, February 10, 2014

RWA#1: Unit M Concept 5 - Conic Sections in Real Life (Parabola)

1. Definition: The set of all points that are equal distance from a point (focus) and a line (directrix)

2. Algebraically, the standard equation is (x-h)^2=4p(y-k) or (y-k)^2=4p(x-h), depending if the graph goes right, left, up, or down. The parabola will go up if x appears first and p is positive. It will go down if x appears first and p is negative. It will go right if y is first and p is positive, and left is when y appears first and p is negative. Your vertex is (h,k) and the directrix is a dotted line that is p units away from the vertex, in the opposite direction of the focus. The axis of symmetry will be perpendicular to the directrix and it will always go through the vertex and the focus. In addition, you don't have to find the eccentricity because it is always be equal to one. 
    Graphically, the parabola goes in four directions depending on the x and y, as mentioned above. The vertex is in between the focus and directrix (FVD or DVF). The focus will always be on the inside of the parabola. As seen on the picture above, the point from the focus to a point on the parabola to the directrix will be the same distance.

Here's a cool video to learn more on parabolas: This video will show you that there are parabolas in video games such as Mario. It shows how the parabola would become skinnier/fatter if he ran while jumping or jumped normally.

3. Real World Application: Not only does parabolas appear in video games (shown in the video above), but they also appear on bridges, roller coasters, and satellite dishes. The picture displayed below is a Roman aqueduct, it would help channel water and was a major component in the Roman water supply. A more sophisticated example would be the satellite dishes. The structure of the dish enables it to transmit and detect signals from a specific source. When the signal bounces off of the dish it creates a parabola.

4. Work Cited: