I/D #1: Unit N: Concept 7: Unit Circle Derivation

INQUIRY ACTIVITY SUMMARY:

The above image shows the whole unit circle with the first quadrant filled out completely with all three Special Right Triangles. The second quadrant shows the 45 degree triangle as expressed with the x-axis (45 degree reference angle). The third quadrant shows the reference angle of the 30 degree angle triangle. The fourth quadrant shows the reference angle of the 60 degree angle triangle.

The above image shows the 30 degree angle triangle. The side opposite the 30 degrees is labeled as "x", the side adjacent is x radical 3 and the hypotenuse is 2x. These shows the lengths of the sides. I then simplified the three sides of the triangle by dividing everything by the length of the hypotenuse. By drawing the triangle as if it were on a coordinate plane with the origin being represented by the 30 degrees point, I found the vertices of the triangle as ordered pairs. These same ordered pairs are used in the unit circle.

The above image shows the 45 degree angle triangle in a similar position as the 30 degree triangle. This time, the length of the hypotenuse is x radical 2 and both the opposite side and the adjacent side are x because the 45 degree triangle has two 45 degree points. Again, the hypotenuse is divided by every side's length. The triangle is "graphed" in order to find the ordered pairs.

The above image likewise shows the 60 degree angle triangle. This triangle is basically exactly the same as the 30 degree angle triangle except that it is shifted so that when graphed, the origin will be the 60 degree point instead. When the hypotenuse is divided by everything and the ordered pairs are found, they correspond with the 30 degree angle triangle.

The above image shows the first quadrant of the Unit Circle that I drew, complete with the triangles and their angles. As long as I know the fist quadrant, the rest of it should come easily because The rest of the quadrants are basically the same, except for a certain shift. The first quadrant gives me the reference angles, and by knowing the reference angles, I will know the ordered pairs that correspond with each degree and depending on which quadrant the angle I'm looking for is on will make my life easier.

The above image shows the whole entire unit circle filled out with the degrees, radians, and ordered pairs once the triangles are applied to it. This activity helped me derive the Unit Circle because I actually got to understand the Unit Circle. I now see the triangles and the reference angles as pertaining to the x-axis. I get the bigger picture of it all.

INQUIRY ACTIVITY REFLECTION:
1. The coolest thing I learned from this activity was there were special right triangles in the unit circle. When I was introduced to the unit circle last year, all that was required was to simply memorize everything on the circle. I didn't realize there were Special Right Triangles in the unit circle nor that that was why everything was arranged. It was that sort of "aha!" moment for me.
2. This activity will help me in this unit because once I figured out that there were Special Right Triangles in the Unit Circle, I was able to understand why things are the way they are. Now, instead of simply memorizing it, I will be able to know why and how it all connects together.
3. Something I never realized before about special right triangles and the unit circle is how very interconnected they are with each other and that triangles could make up a circle.