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

Inquiry Activity Summary 

The activity that we did in class demonstrated to us how special triangles are related to the unit circle. It was review of the special right triangles and the side lengths of the triangle depending on the special right triangles of 30º, 60º, 45º.

30º Triangle! 

A 30º triangle has special qualities. The shortest side opposite of 30º is X. The hypotenuse is two times x and is across the 90º angle. The side across the 60º angle is x times radical three. There are rules to the 30º triangle. To simplify the three sides, the hypotenuse is set to 1. In order to get to 1, we must divide the hypotenuse by itself, making it 2x/2. Thus, r is equal to 1. Since we did it to one side, we must do it to the others. The opposite side (x) is radical 3 over two because we have x over 2x and the x's cancel out, leaving you with 1/2.  With the adjacent side, you do x radical 3 over 2x. The x's cancel out leaving you with radical 3/2. When you draw it on a coordinate plane, you think of your adjacent as your x axis and opposite as your y axis.

45º Triangle! 

A 45º triangle is special because there are two angles that are equivalent. That means that both of their opposite sides are the same as well. The hypotenuse is x radical 2. For a 45º, we divide the hypotenuse by x times radical 2, so the hypotenuse becomes 1. We divide x by x radical 2 to the other two sides. The x's cancel out and when you rationalize, it simplifies to radical two over two. Therefore, the y value is radical 2 over 2 because you are going up radical 2 over 2.

60º Triangle!

A 60º triangle has the same characteristics as 30º, as in the sides. Thus, we divide the hypotenuse by itself to get it to 1. Though it is similar to the 30º,  the triangle's visual is different, as in the orientation. This changes the x coordinate in the adjacent side to 1/2. The opposite side length (vertical) coordinate will become radical 3 over 2 for the y coordinate. 

Here is a video that clarifies how I find the sides of each triangle. 

How does this activity help you derive the Unit Circle? 

Once you draw a plane for the triangles, it is visible that the new triangles are in the first quadrant. This shows how the ordered pairs work in the Unit Circle. Since the hypotenuse is equal to 1, that makes the radis of the Unit Circle 1. Thus, with the origin being at (0,0), we soon see the side lengths as coordinates that we use in the Unit Circle. We also see that every quadrant in the plane looks the same, just the difference is the negative values in either x or y in II, III, and IV. But, with the triangle drawn in quadrant 1, we see that it reflects onto the other quadrants, which creates the same reference angles, which is key to completing the Unit Circle.  The point at 0 degrees is (1,0), 90 degrees is (0,1), 180 degrees is (-1,0), and 270 degrees is (0, -1) because the radius is 

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?

The triangle in this activity lies in the first quadrant. The values change from quadrant to quadrant because they share a common origin and its just a reflection of the triangle in each quadrant.

30º 

(http://01.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_34.gif)

If you reflect a 30º triangle to the 4th quadrant, you see that the for the side across the 30º angle changes. Instead of being a positive on both the x and y coordinates as seen in the first quadrant, the y value is -1/2. This is because in the fourth quadrant, the y value is always negative. Therefore, the coordinate is (radical 3 over 2, -1/2). The adjacent angle (across 60º) stays the same because it is not being affected as it reflects. 

60º 

(http://02.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_21.gif)

If you reflect a 60º triangle in the third quadrant, both values are negative. As you can see, everything stays the same as in value, but it becomes negative because it is in the third quadrant. The fourth quadrant is similar, the x value is positive and the y value is negative. 

45º 

(http://01.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_45.gif)

If you reflect a 45º triangle to the second quadrant, the coordinates remain the same. The only difference is that the x value is negative. This is because the second quadrant always contains a negative x value. 

INQUIRY ACTIVITY REFLECTION 

1. The coolest thing I learned from this activity was that the Unit Circle is not just memorization, but there is a meaning behind it all which makes everything so much simpler.
2. This activity will help me in this unit because last year in Algebra II I didn't understand the unit circle and I struggled a lot, mainly because I didn't understand the triangles. Now, I can solve the unit circle because I know why it works the way it does!
3. Something I never realized before about special right triangles and the unit circle is that THEY ARE RELATED! It helps so much because now I don't have to convert degrees to radians!

(p.s.) Sorry for the white background shadowing for the words, I don't know how to fix it! If you could teach me during class one day it would be great!)

References:

1. (http://01.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_34.gif)

2. (http://02.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_21.gif)

3. (http://01.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_45.gif)

RWA# 1: Unit M Concept 5: Graphing Ellipses Given Equation

1. The mathematical definition of an ellipse is "the set of all points such that the sum of the distance of two points, known as the foci, is a constant" (Kirch).
2.     The standard formula of an ellipse is (x-h)^2/a + (y-k)^2/b = 1. On a graph, an ellipse is depicted as an oval, or a squashed circle. In other words, it is stretched out. The key features of an ellipse are the standard form, the center, vertices, co-vertices, two foci, major axis, minor axis, a, b, and c, eccentricity,  and if it is seen as "skinny" or "fat". You can find these graphically or algebraically.
         Lets start off with algebraically. By looking at the standard form of an ellipse, you can easily find the center. The x value stands for "h" and the y value stand for "k". After you find your center, you can find your major axis by looking at the denominator. If the denominator is bigger under the first term (the x), that is your major axis. This also means that your ellipse will stretch horizontally, being "fat". If the bigger denominator is under the y, then then your graph will stretch vertically, being "skinny". Once you have your major axis, you can find your minor axis because it would be opposite. The smaller number x is your minor axis and horizontal. If the smaller number is under y then it is vertical.  For major and minor, if one is vertical, you know the equation is x= h is. If it is horizontal, y=k.You can find your a and b from the standard formula too. The bigger denominator is the a^2 term so you have to square root it to find your a. The smaller denominator is your b^2 term and you also have to square root it to find your b. After that, you can find your c term by plugging your "a" and "b" into the formula, a^2+b^2=c^2. To find the vertices, you add and subtract "a" from your center. The major axis stays the same throughout. To find the co-vertices, you add and subtract "b" from your center, the minor axis staying the same. To find the foci, you add "c" to the value that's changing. It's on the major axis. If major axis is x=, you add "c" to your "k" (y value of center). It is plus or minus because there are two vertices. If the foci is closer to the center, the ellipse is more circular. If it is farther, it is more stretched out. Eccentricity is  a measure of how much the conic section deviates from being circular. If it is closer to 1, is it more stretched out. If it is closer to 0, it is more in the circular form. The eccentricity is between 0 and 1. To find the eccentricity, you use the formula c/a and round to the thousand's place.
             To identify an ellipse graphically, it is simple as well. The major axis is the longer diameter of the ellipse, and is seen with a straight line (no dashes). The minor axis is the axis intersecting the major axis, it is shorter, and seen with a dashed line. The vertices are the points that lie on the major axis. The co-vertice are the points that connect the minor axis, at the end. With the vertices and co-vertices, the ellipse can be drawn. The center of the ellipse is where the major and minor axis intersect. If the ellipse is seen more horizontally, then it is "fat". If it is seen as more vertically stretched, it is "skinny" You can determine a by counting how many points are away from the center from each vertices on the major axis. Same goes with how to find b, you can count how many points are away from the center of the co-vertices on the minor axis. The foci is within the vertices of the major axis. The closer the foci to the center, the more it is in circular form. The farther the foci, the more it deviates from being a circle. If the foci is close to the center, the eccentricity is close to 0. If foci is far from center, eccentricity is big.

Here is a website that shows how to graph an ellipse!
http://www.purplemath.com/modules/ellipse2.htm

http://www.mathwarehouse.com/ellipse/equation-of-ellipse.php

Here is an ideal graph of an ellipse with some of the points I talked about. 

This is a cool video on how to find the foci! 

3.       A real world application of an ellipse is that it is used in lithotripsy. "One important property of the ellipse is its reflective property. If you think of an ellipse as being made from a reflective material then a light ray emitted from one focus will reflect off the ellipse and pass through the second focus. This is also true not only for light rays, but also for other forms of energy, including shockwaves. Shockwaves generated at one focus will reflect off the ellipse and pass through the second focus."(http://mathcentral.uregina.ca/beyond/articles/Lithotripsy/lithotripsy1.html). 
           Therefore, it has been used my medical experts to create a device that helps kidney stones and gallstones. A lithtripter uses shockwaves to SHATTER a kidney stone or gallstone. Patients with kidney stones or gallstones can be saved because of the shape of an ellipse because it reflects shockwaves. HOW COOL IS THAT?! MATH SAVES LIVES AND IT WILL CONTINUE TO. :D 

4. References:
Kirch
http://mathcentral.uregina.ca/beyond/articles/Lithotripsy/lithotripsy1.html
http://www.youtube.com/watch?v=3O_TMiP9piI
http://www.purplemath.com/modules/ellipse2.htm
http://www.mathwarehouse.com/ellipse/equation-of-ellipse.php