I/D #1: Unit N: Concept 7: Unit Circle & Its Relation to Special Rights Triangle

INQUIRY ACTIVITY SUMMARY

In class, Mrs. Kirch gave us a worksheet which had pictures of 30*, 45*, and 60* triangles.

First we labeled the triangle following the given rules of the Special Right Triangle. The hypotenuse (r) is 2x, (y) is x, and (x) is x radical 3. After that, we had to make the hypotnuse equal to 1. Since the hypotnuse is 2x, we divided all 3 sides by 2x. R simplified to 1. In Y, the x's canceled and simplified to 1/2. In x, radical three/2x, the x's canceled out, simplifying to radical 3/2. We then drew the coordinate plane with the triangle lying in quadrant 1, making the origin (0,0). Since a point consists of (x,y) we substitued x for radical 3/2, and substituted 1/2 for y (highlighted in blue). 

   Following the same steps, we then solved for the 45* triangle. The hypotenuse (r) is x radical 2, and both (y) and (x) are x. We continued to make the hypotnuse equal 1. Since the hypotnuse is x radical 2, we had to divide all 3 sides by x radical 2. We divided x by x radical 2 but since we can not have a radical as the denominator, we rationalized by multiplying radical 2 on both top and bottom which simplified to radical 2/2. After, we drew a coordinate place with the origin being (0,0).  And since we know an ordered pair is (x,y) we get the ordered pair of (radical 2/2, radical 2/2) (highlighted in green).

 The last angle we solved was the 60* triangle. We followed the same steps as before. The hypotenuse (r) is 2x, (y) is x radical 3 and (x) is x. Then, we have to make the hypotnuse equal 1. Since the hypotnuse is 2x, we have to divide all 3 sides y 2x. r simplified to 1, (y)'s x radical 3/2x simplified to radical 3/2, (x)'s x will simplify to be 1/2. We drew the coordinate plane setting the origin to (0,0). Knowing that an ordered pair is (x,y), the 60* triangles ordered pai is (1/2, radical 3/2) (highlighted in green).

This activity helped me derive the ordered pairs for the unit circle, using the 30*, 45* and 60* triangles. 

The quadrants which I showed lie on the first quadrant. The number of the values do not change if we put it in Quadrant II, Quadrant III or in Quadrant IV, only the signs change.

As you can see highlighted in this picture, the signs change varied of what quadrant the triangle is in.

The coolest thing I learned from this activity was... learning that the unit circle is not a bunch of random numbers I had to memorize, it actually makes sense knowing why/how the ordered pairs are there.

This activity will help me in this unit because… now that I know the magic 5, which are located in the 1st quadrant, I know the rest of the ordered pairs.

Something I never realized before about special right triangles and the unit circle is… that the special triangles overlapped each other. Wait, I did not even know there were triangles in the unit circle.

RWA# 1: Unit M Concept 4-6 : Conic sections in real life

Ellipses

1) The mathematical definition of an ellipse is, "The set of all points, such that the sum of the distance from 2 points known as the oci, is a constant." -Mrs. Kirch

2) An ellipse includes the center, major and minor axis, 2 verticies, 2 co-verticies, 2 foci and and eccentricity along with a, b and c.

Here is what the equation of an ellipse looks like: 

(http://www.descarta2d.com/BookHTML/Chapters/ell.html)

Here is how an ellipse looks like graphically: 

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

We can automatically tell how our graph is going to look like graphically when the equation is given. We can tell if the ellipse is "fat" when the biggest number comes first(under x) and the ellipse will be skinny if the bigger number is under y. We can also figure out the center by using (x,y) and plugging it into (h,k) when x goes with h, and y goes with k. Both verticies will be the same number away from the center, along with the co-verticies and the 2 foci. 

To figure out how the equation will graph, we must find some "hints" that can lead to other answers. First, once the equation is given, we can find out the center, which is (h,k). After finding out the center, we can find out "a" and "b". The square root of "a" will always be the bigger number when dealing with ellipses and the smaller number will be "b". To figure out c, we can use "a^2-b^2=c^2. Once c is found, we can find the eccentricity which would be "c/a". The eccentricity of an ellipse is between 0 to 1. The verticies, co-verticies, foci, major and minor axis depend on whether the bigger number is under x or y.  

For further explanation, refer to the video below. 

3) Below, there is a picture of a soccer/track field in the shape of an ellipse. The midpoint would be the point in the middle which separates the middle of the field. The major axis would go horizontally, and the minor axis would go vertically. In this case, algebraically, the bigger number would come first, under x.

(https://www.google.com/search?q=racetrack&safe=active&client=firefox-a&hs=hai&rls=org.mozilla:en-US:official&source=lnms&tbm=isch&sa=X&ei=46r6UsiUH6O92wW4poC4Cw&ved=0CAoQ_AUoAg&biw=1280&bih=647#facrc=_&imgdii=_&imgrc=bWTXj9M8I81LGM%253A%3BV553ddJQAu7xVM%3Bhttp%253A%252F%252Fus.123rf.com%252F400wm%252F400%252F400%252Fexperimental%252Fexperimental1001%252Fexperimental100100014%252F6252442-football-soccer-field-pitch-vector-along-with-racetrack.jpg%3Bhttp%253A%252F%252Fwww.123rf.com%252Fphoto_6252442_football-soccer-field-pitch-vector-along-with-racetrack.html%3B1200%3B936)

4) Work Cited: 

(http://www.descarta2d.com/BookHTML/Chapters/ell.html)

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

((https://www.google.com/search?q=racetrack&safe=active&client=firefox-a&hs=hai&rls=org.mozilla:en-US:official&source=lnms&tbm=isch&sa=X&ei=46r6UsiUH6O92wW4poC4Cw&ved=0CAoQ_AUoAg&biw=1280&bih=647#facrc=_&imgdii=_&imgrc=bWTXj9M8I81LGM%253A%3BV553ddJQAu7xVM%3Bhttp%253A%252F%252Fus.123rf.com%252F400wm%252F400%252F400%252Fexperimental%252Fexperimental1001%252Fexperimental100100014%252F6252442-football-soccer-field-pitch-vector-along-with-racetrack.jpg%3Bhttp%253A%252F%252Fwww.123rf.com%252Fphoto_6252442_football-soccer-field-pitch-vector-along-with-racetrack.html%3B1200%3B936)
https://www.youtube.com/watch?v=5nxT6LQhXLM