I/D#1:Unit N Concept 7: Derive the Unit Circle Activity

Inquiry Activity Summary:

In this activity we were given 3 triangles which had the measurements of: 30, 60,90, 45,45,90 and 60,30, and 90 degrees. These measurements are of Special Right Triangles. We had to label each triangle according to the rules of Special Triangles. Which are:

http://www.math.hmc.edu/calculus/tutorials/reviewtriglogexp/Add caption

The hypotenuse of each triangle had to equal 1. The first triangle was a 30 degree which meant that the sides where labeled as 2x, x radical 3, and x. To make the hypotenuse 1, we had to dive all sides by the hypotenuse and simplify. After that we labeled the hypotenuse "r", horizontal value"x", and vertical value "y". The next step is to draw a coordinate(this has to be done to very triangle given) with the origin at the labeled measure, which for the first one was 30 degrees. The vertices had to be labeled as ordered pairs for each triangle. For the first triangle, which is 30 degrees, The hypotenuse equal 2x(r), and the sides were x(vertical value), and x radical 3(horizontal value). To simplify I had to divide all sides by 2x, which gave me 1 for the r value, radical 3 over 2 for x, and 1/2 for y. I later drew the coordinate and labeled the order pairs which were (0,0), (radical 3/2,0), and (radical 3/2, 1/2). That is how the ordered pairs that are in the Unit Circle came to be for any reference angle of 30. The pairs are the same for any reference angle of 30.
1)30 Degree Triangle:

2) 45 Degree Triangle:
For the 45 degree angle, the hypotenuse was x radical 2(R), vertical side was x(x), and horizontal side was x(Y). For this triangle we had to divide by radical 2, which gave R=1, X= radical 2/2, and Y= radical2/2. After drawing the coordinate plane, the vertices were: (0,0), (radical 2 over 2, 0), and (radical 2 over 2, radical 2 over 2). That is how the ordered pairs for the quadrants in the unit circle came from, for angles that were reference angles of 45 degrees.

3) 60 Degree Triangle:
For the 60 degree triangle, the hypotenuse was 2x(R), horizontal side was x(x), and vertical side was x radical 3(y). We had to divide by 2x which gave r=1, x=1/2, and y= radical 3 over 2. After drawing the coordinate plane, the vertices were (0,0), (1/2,0), and (1/2, radical 3 over 2). That is how the pairs for any reference angle of 60 came to be. For any reference angle of 60 in the unit circle the pairs will be the same.

4) This activity helps us obtain the unit circle because the ordered pairs that we got for the tree triangles are the same in each quadrant of the unit circle, which means that if you know the pairs for the 30, 45, and 60 degree angles you will know the complete unit circle. As each of the tree angles have reference angles in each quadrant.

5)The triangles drawn lie in the first quadrant, which makes the ordered pairs positive. If the triangles were drawn in different quadrants the pairs would change to negative depending in what quadrant they are. After re-drawing the triangles, for the 30 degree triangle in the second quadrant, the x values of the ordered pair became negative. For the 45 degree triangle the x and y values both became negative in the third quadrant. For the 60 degree triangle the y values became negative when drawn in the fourth quadrant.

Inquiry Activity Reflection:

1. "The coolest thing I learned from this activity was" where the unit circle came from.
2. "This activity will help me in this unit because" it will help be get reference angles faster and the ordered pairs.
3. "Something I never realized about special right triangles and the unit circle was" that they had so much in common or that the triangles were used to make the unit circle.

RWA#1: Unit M Concept 5: Graphing ellipses given equation

  

 Section 1:

• The set off all points, such that the sum of the distance known as the foci, is constant.
• Equation: (x-h)^2 / a^2 + (y-k)^2 / b^2=1 or (x-h)^2 / b^2 + (y-k)^2 / a^2=1 and a^2-b^2=c^2
• The key points of an ellipse are: the center, a, b, c, 2 vertices, the major axis, 2 co-vertices, the minor axis, 2 foci, and the eccentricity. To find A and B is the standard form is given A will always be the bigger number. Depending if x or y come first the graph will be either skinny or fat. If Its skinny the x value will not change from the center. to get the vertices, if its skinny the x values wont change and you will need to add and subtract whatever number a  is to find the y values. For the co-vertices the y will be the same as the center and subtract and add whatever number b is to find the x values. The major axis will depend on if its skinny or fat, if its skinny major axis will be x= whatever number x is for the center and the minor axis will equal whatever y is in the center. For a fat graph the numbers will change y will become major and x minor. To find the foci you need to know c and depending if its skinny or fat the x or y values will change from the numbers that make up the center. To find C you use  a^2-b^2 =c^2. The eccentricity will be C over A.

The closer the foci are to the vertices the ellipse will become fat and the closer they are to the center it will become more narrow or skinny.


Section 3:
A real world application that displays ellipses are earrings. Earrings can be worn on the ears and can be different material and color. Earrings can be in the form of ellipses.Most women over the world use earring. This real world application contains different sizes for ellipses.

Section 4:
http://www.mathamazement.com/Lessons/Pre-Calculus/09_Conic-Sections-and-Analytic-Geometry/ellipse.html
http://official-stardollfashion.blogspot.com/2009/06/8-hottest-trends-for-summer-2009.html