Inquiry Activity Summary:
Angles on the first quadrant
30 degree angle
Figuring out the 30 degree angle triangle was very simple, though at first it was a bit difficult. The reason being because I didn't know the measurements of each side. But once I logged onto legoogle.com, I was able to find all the sides. Side R = 2x, side Y = x and side X = xradical3 (as seen in the picture.) Our hypotenuse/sideR had to equal one, therefore, we divided out 2x from the 2x and we got side R to equal 1. Simple right, are you done yet? Nope. In order to divide out the 2x from sideR we had to do that to all the sides. SideX when we divided and simplified it gave us radical3 over two. For sideY we did the same and the x's cancelled therefore leaving us with 1/2.
The second part of the 30 degree angle was finding the points as if it were on a graph. Taking that initiative we drew out the x and y axis. Our first point on the graph was the center point and that was simply just (0,0). The second point on the the x-axis, on the right, would turn out to be (radical3/2, 0) because of the information we had solved earlier. According to rise over run, our last point would have to be (radical3/2, 1/2). Why? Because we went over radical3/2 and rose 1/2. Lastly we filled in the blanks for r, x, and y.
45 degree angle
Figuring out the 45 degree angle was much simpler than the first angle we had to figure out because we already had an idea of how to solve it out. According to our sources, google.com, sideR was xradical2, sideX and sideY were both x. Just as we did for angle 30, sideR had to equal 1. Therefore, we divided out xradical2 from xradical2 to get one. We divided out xradical2 from all the sides. When dividing out xradical2 from x we ended up with 1/radical2, of course we all know, that no radical can be in the denominator, so we multiplied radical2 to both the bottom and top. Our final result was radical2/2. From there we drew the x and y axis and figured out the points, resulting with what is above in the picture. Lastly, we filled in the r, x and y blanks.
60 degree angle
While we all solved this angle we slowly realized that everything was the exact same as the 30 degree angle except everything was reversed. SideR was the same. But the X for the 60 degree angle was 1/2 instead of being radical3/2 and sideY was radical3/2 instead of 1/2. As for the points as well! Instead of going through all the work, since we already knew how to do it from the previous 30 degree angle, we just reversed everything and finished our 60 degree angle in seconds.
How did the activity help us derive the Unit Circle?
Well, glad you asked. These three angles are key to finding the rest of the Unit Circle. All we need to know is each and every angle (referring to 30, 45, 60 degree angle) and we can find the rest of the unit circle.
http://fac-web.spsu.edu/math/edwards/1113/unitcircle.htm
As seen from the Unit Circle above, the only points that are repeatedly seen are 1/2, radical3/2 and radical2/2. That's because all angles have the same common thing, they all have reference angles of 30, 45, and 60. 30 and 150 reflect off one another and have the same point except for the negative x, but that makes sense since it is on the negative side of the x-axis. This method goes to the rest of the angles as well they all reflect one another and tie back to either a 30, 45 or 60 degree angle. The only thing that changes throughout the Unit Circle is the degrees and radians but that's simply because as we go around the circle the angle gets bigger and bigger.
Quadrant II (30 degree angle)
This angle above in fact is the 150 degree angle. But it has the reference angle of 30 degrees. The only thing that really changes compared to the original 30 degree angle is that the 150 degree angle has a negative radical3/2 and of course the degree and radian are different as well.
Quadrant III (45 degree angle)
The angle above is actually 225 degrees. This angle has a reference angle of 45 degrees. The only thing that really changes compared to the original 45 degree angle is that the 225 degree angle's the x and y points are negative. To conclude, the degree and radian are one of the other factors that are different.
Quadrant IV (60 degree angle)
The angle above is actually 300 degrees. This angle has a reference angle of 60 degrees. The only thing that really changes compared to the original 60 degree angle is that the 300 degree angle's x and y points are negative. Lastly, the degree and radian are one of the other factors that are different.
Inquiry Activity Reflection:
The coolest thing I learned from this activity was that I finally understood how to solve the Unit Circle! Last year, in Algebra II, when we were learning this, I was so lost. I wanted to love the Unit Circle but the Unit Circle didn't love me. It was a very depressing time for me. But now me and the Unit Circle are on the same terms. I get it and the circle gets me. It's a happy world isn't it? Yes, very.
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