The activity we did in class basically used our knowledge of geometry to derive the Unit circle from Special Right Triangles of 30, 45, and 60 degrees. We basically had to set our hypotenuse equal to zero. Then, we had to do the same to the other two legs (simplifying the triangle legs fairly). Then, we labeled the triangle with the hypotenuse being r, the vertical leg being y, and the horizontal leg being x, and then placed the triangles in a coordinate plane with the the origin located at the angle measured ( to get the triangle in Quadrant 1. Finally, we labeled the vertices and found the ordered pairs.
1)
This triangle above is the 30 degree special right triangle. The leg opposite to 30° is the y value (vertical), the leg opposite to 60° is the horizontal value (horizontal), and the leg opposite to the 90° angle is the r value (hypotenuse). In order to find our ordered pairs, we had to set the hypotenuse equal to 1. In other words, we had to divide by 2x. If you divide 2x from the hypotenuse, then you need to divide all of them by 2x. When you do this you are left with x=√3∕2, y=1∕2, and r=1. The ordered pair is (√3∕2 , 1∕2)
2)
The triangle above is the 45° special right triangle. The leg opposite to the lower 45° angle is the y value (vertically), the leg opposite to the higher 45° angle is the x value (horizontal), and the leg opposite to the 90° angle is the r value (hypotenuse). Once again, we need the hypotenuse to be equal to 1, so we need to divide by √2x. When we do this all legs, the all the Xs cancel out, but you are left with radicals on the bottom so you need to rationalize. In order to do this, you multiply √2 to both the top and the bottom. When you multiply the bottom, the radicals cancel out, and you are left with √2∕2. This means that the values end up being x=√2∕2, y=√2∕2, and r=1. The ordered pair is (√2∕2 , √2∕2)
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| http://upload.wikimedia.org/wikipedia/commons/1/15/Triangle_30-60-90_rotated.png |
2)
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| http://mathcountsnotes.blogspot.com/2012/05/special-right-triangles-30-60-90-and-45.html |
3)
This triangle above is the 60 degree special right triangle. The leg opposite to 60° is the y value (vertical), the leg opposite to 30° is the horizontal value (horizontal), and the leg opposite to the 90° angle is the r value (hypotenuse). In order to find our ordered pairs, we had to set the hypotenuse equal to 1. In other words, we had to divide by 2x. If you divide 2x from the hypotenuse, then you need to divide all of them by 2x. When you do this you are left with x=1∕2, y=√3∕2, and r=1. The ordered pair is (1∕2, √3∕2)
4) This activity helps us derive the unit circle because these are basically the three triangles found in the unit circle. All that you need to do is move the triangles around into the other quadrants in order to find the rest of the unit circle.
5) All the triangles in this activity lie in Quadrant I. If we were to transfer the triangles into the other Quadrants, only the signs of either the X or Y values would change. In other words, instead of having a positive X, you would have a negative X, and so on.
In this triangle we see that the ordered pair has remained almost the same for the 45° angle. The only thing that has changed is that the ordered pair is no longer (x , y) but rather (-x , y) due to being located in Quadrant II.
In this triangle we see that the ordered pair has once again changed slightly from original form for the 30° angle. Instead of having an ordered pair whose signs are (x , y), we have and ordered pair whose signs are now (-x , -y) due to being in located in Quadrant III.
In this triangle we see that the ordered pair has (like in the last two) changed slightly for the 60° angle. instead of having the ordered pair of (x , y), we have the ordered pair being (x, -y) due to now being located in Quadrant IV.
Inquiry Activity Reflection:
1) The coolest thing I learned from this activity was the Special Right Triangles. I personally do not remember learning this or the Unit Circle in my past two years of taking math.
2) This activity will help me in this unit because I can draw out the triangles in Quadrant I, label the ordered pairs, and write out the "ALL STUDENTS TAKE CALCULUS" and I'll have the whole Unit circle.
3) Something I never realized before about Special Right Triangles and the Unit Circle was that they existed.
WORK CITED:
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| http://mathcountsnotes.blogspot.com/2012/05/special-right-triangles-30-60-90-and-45.html |
4) This activity helps us derive the unit circle because these are basically the three triangles found in the unit circle. All that you need to do is move the triangles around into the other quadrants in order to find the rest of the unit circle.
5) All the triangles in this activity lie in Quadrant I. If we were to transfer the triangles into the other Quadrants, only the signs of either the X or Y values would change. In other words, instead of having a positive X, you would have a negative X, and so on.
 |
| http://01.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_45.gif |
In this triangle we see that the ordered pair has remained almost the same for the 45° angle. The only thing that has changed is that the ordered pair is no longer (x , y) but rather (-x , y) due to being located in Quadrant II.
 |
| http://01.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_34.gif |
In this triangle we see that the ordered pair has once again changed slightly from original form for the 30° angle. Instead of having an ordered pair whose signs are (x , y), we have and ordered pair whose signs are now (-x , -y) due to being in located in Quadrant III.
 |
| http://02.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_21.gif |
In this triangle we see that the ordered pair has (like in the last two) changed slightly for the 60° angle. instead of having the ordered pair of (x , y), we have the ordered pair being (x, -y) due to now being located in Quadrant IV.
Inquiry Activity Reflection:
1) The coolest thing I learned from this activity was the Special Right Triangles. I personally do not remember learning this or the Unit Circle in my past two years of taking math.
2) This activity will help me in this unit because I can draw out the triangles in Quadrant I, label the ordered pairs, and write out the "ALL STUDENTS TAKE CALCULUS" and I'll have the whole Unit circle.
3) Something I never realized before about Special Right Triangles and the Unit Circle was that they existed.
WORK CITED:
http://upload.wikimedia.org/wikipedia/commons/1/15/Triangle_30-60-90_rotated.png
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