Inquiry Activity Summary
#1: The 30º Triangle
In this class activity, we learned where and how to get the plotted points of the Unit Circle's Magic three triangles located on the 1st Quadrant. The triangles are the (30,45,60 degrees) by using the Rules of Special Right Triangles. To start the activity we first are required to label all three sides.This includes getting the Hypotenuse to equal R, the Horizontal value to be X and the Vertical Value to be Y. In this case, the Hypotenuse is equal to one as you can see in the picture we have 2x/2x (as instructed) which gives us our answer of 1 since we can cancel out the 2's leaving us an answer of 1. Then we have to find the other two sides. Which takes us back to knowing the special rules that apply to this particular triangle. This that the x side is x-radical 3 and the y side is just x. Since we since we divided by 2x in the hypotenuse, we are obliged do the same through out the triangle and then simplify. So x radical 3 over 2x turns to radical 3/2 which is your x value since we can cancel the x values.. Then we do the same for our y value and x/2x turns to 1/2, again by canceling the value of x. Then we draw a graph on the triangle starting from (0,0). So were the 30º opens up will be (0,0), where the 90º open up will be (radical3/2,0) and where the 60º opens up, the value will be (radical3/2,1/2). The whole point of this activity is supposed to make you realize that the unit circle is just the 3 "magical triangles" that repeat it self in different quadrants. So instead of viewing it as a circle, you view it as three triangles.
#2: The 45º Triangle
Now, on to the 45º triangle, First we do the same basic concept. Label the sides accordingly to the special rules of the 45º triangle Since it is a 45 degree triangle it varies just a little bit. This means that even though the hypotenuse still equals to 1, we must now divide by radical 2 in order to get it to 1. Since this is a 45º triangle, two side are the same so that means both the horizontal and vertical value is equal x. When you divide by a radical 2, you take out the x and leave it as 1/radical 2. Once you have that you must multiply by radical 2 on top and bottom. THERE CAN BE NO RADICAL IN THE DENOMINATOR. So you end up with radical 2/2 . Again we do the graph that connects to get 45º to equal (0,0), the 90º to equal (radical2/2,0) and the second 45º to be (radical2/2,radical2/2). This again relates to the unit circle because as you go into different quadrants, you're going to notice that they're the exact same points just the (x,y) have a negative/positive sign.
#3: The 60º triangle
The 60º triangle is very similar to the 30º one. It uses the same rules its just switched around. The Horizontal value is now x and the vertical value is now x-radical 3. With this, we use the exact proceedures as before making 2x/2x=1 (for our hypotenuse) so we divide each side by 2x. If you notice, its the same exact thing with a 30 degree triangle so you basically did the work already (yayy! {=~^0^~=}) However as i mentioned before, it is flipped so our x value is now 1/2 and our y value is now radical3/2. Once you do the graph you realize it is the same points as a 30 degree triangle (but the x and y are just flipped). so where 60º opens up is (0,0), where 90º opens up is (1/2,0), and where 30º opens up is now (1/2,radical3/2).
4. How does this help derive the unit circle?
This helps me because instead of trying to kill yourself learning radical this radical that, (which i have been doing with great success but way to much time), you are able to use just 3 different triangles and simply change whether the x is +/- and if the y is +/- which can be easily determined by the quadrant. Plus it makes it more fun to understand that every unit has all these other angles, and sides and with just one quadrant, you know the whole thing!
5. How do Quadrants affect the values?
Simple. It all depends on each quadrant, since each quadrant is like graphing a simple point. If the the triangle lies within that specific quadrant, for example; Quadrant II, than the the values will be (-x,+y), if it lies in Quadrant III, then the values will change to (-x,-y). And if they lie on Quadrant IV, then the values change to (x,-y).
Inquiry Activity Reflection
A: The hardest thing was actually the most fun and that's that I've learned from this activity where all the points in the Unit circle came from. This was especially important because last year as a sophomore, I had surgery and was absent from school for about a month. So i never learned the UNit Circle, or Logs or anything of second semester, so to learn something like this without prior knowledge is important to me c': . But now, as a junior in Math-analysis, it all makes so much sense rather than just being scared of weird numbers x)
B: This activity will definitely help me in this units test because, i can simply refer to the Special Right Triangle rules to help me remember anyhting i forgot and finish the unit circle and as long as i can complete quadrant I, i can complete them all.
C: Something I never realized before about Special Right Triangle and the Unit Circle is that they were both related to each other. Plus I didn't know they existed to be honest. When I first heard of this activity i was a bit scared to fail because i'm not the best at math so for me to remember all these radicals, and numbers, and degrees it felt like so much >_< but with practice and constant support from the teacher, i was able to figure them out and feel confident in myself and my performance on my test.
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