I/D# 1: Unit N Concepts 7-9 : How Do Special Right Triangles and the Unit Circle Relate?

INQUIRY ACTIVITY SUMMARY
1. Describe a 30 degree Angle 
      I first labeled the triangle according to the Special Right Triangle Rules which is the hypotenuse 2x, the horizontal value x radical 3, and the vertical value x. I then had to simplify the three sides of the triangle fairly in a way such that the hypotenuse=1 so i divided each side by 2x (which is the hypotenuse) giving me: r=1, x=radical 3 over 2, and y= 1/2. I labeled the hypotenuse r, the horizontal value x, and the vertical value y. In order to draw a coordinate plane for each angle i first put the ordered pair (0,0) at the 30 degree corner, then since our x= radical 3 over 2, i knew my ordered pair for the 90 degree angle was (radical 3 over 2,0) because it didn't move up the y axis. For the 60 degree angle, i simply already knew that x=radical 3 over 2 and our y=1/2 so i put them as the ordered pair (radical 3 over 2, 1/2) since it moved x units to the right and y units up.

2. Describe a 45 Degree Angle 
      I first labeled the triangle according to Special Right Triangle Rules which is the hypotenuse x radical 2, the horizontal value x, and the vertical value x as well. In order to have the hypotenuse equal to one as  i divided each side by the value of the hypotenuse which is x radical 2 resulting in as r=1, x=radical 2 over 2, y=radical 2 over 2. I labeled the hypotenuse r, the horizontal value x, and the vertical value y. To find the coordinate planes, i applied the same concept as i did for the 30 degree angle above. I put the ordered pair (0,0) at the 45 degree left corner, then since our x= radical 2 over 2, i knew my ordered pair for the 90 degree angle was (radical 2 over 2 over 2,0) because we didn't move up the y axis. For the other 45 degree angle, i simply already knew that x=radical 2 over 2 and our y=radical 2 over 2, so i put them as the ordered pair (radical 2 over 2, radical 2 over 2) because we moved x units to the right and y units up.

3. Describe a 60 degree Angle
      I first labeled the triangle according to Special Right Triangle Rules which is the hypotenuse 2x, the horizontal value x, and the vertical value x radical 3. To have the hypotenuse equaling to one, i divided it by itself and the other sides as well having r=1, x=1/2, y=radical 3 over 2.  I labeled the hypotenuse r, the horizontal value x, and the vertical value y.To find the coordinate planes, i applied the same concept as i did for the above angles. I put the ordered pair (0,0) at the 60 degree left corner, then since our x= 1/2 , i knew my ordered pair for the 90 degree angle was (radical 2 over 2 over 2,0) because we didn't move up the y axis and moved x units to the right. For the 30 degree angle, i simply already knew that x=1/2 and our y= radical 3 over 2, so i put them as the ordered pair (radical 2 over 2, radical 2 over 2) because we moved x units to the right and y units up.

4. This activity helped me derive the unit circle because it gave me the coordinates, points, and angles that are within the unit circle as seen below.  I understood the foundation of it, allowing me to not only have to memorize numbers but know why those numbers exist. I thought it was important to do this activity because it gave me the basic comprehension of what a unit circle is.

5. The 30, 45, and 60 degree angles are within the first quadrant of the unit circle. The second quadrant is a reflection of the first, the third is a reflection of the second, and the fourth is a reflection of the third. In the first quadrant, all coordinates are positive, in the second quadrant, only the x value is negative, in the third quadrant both x and y are negative and in the fourth quadrant only the y is negative having that the first and third quadrant are positive and the second and fourth are negative. For a better understanding look below.

Inquiry Reflection Activity
1. The coolest thing I learned from this activity was how each quadrant is reflected with triangles.
2. This activity will help me in this unit because understanding and memozing the unit circle is a basic need to solve problems in all the concepts.
3. Something I never realized before about special right triangles and the unit circle is that the whole unit circle consists of many right right triangles that give it its coordinates.