I/D #1: Unit N: Concept 7: How do special right triangles and the unit circle compare?

1. The 30* Triangle

The 30* triangle has three different sides to it: adjacent which is x, opposite which is y, and hypotenuse which is r. The side opposite the hypotenuse will always be x.The hypotenuse must be one if you want to derive the unit circle from the triangle. If you want this to happen you have to divide each side by 2x. Once you do this you will get x=radical 3 divided by 2, y=1/2 and r=1. We can use these simplified values as coordinates to determine where 30 degrees lies in a quadrant on the unit circle. We know that 30 degrees on the unit circle is located on radical 3/2, 1/2. This can be used for 150 degrees, 210 degrees, and 330 degrees. The only difference is that they are located on different quadrants and there will be negatives and positives.

2. The 45* Triangle

The 45* triangle has two sides that are the same length which are x and y the hypotenuse is r. To derive the unit circle from the triangle we have to get the hypotenuse to equal 1. In order to do this we divide every side by  x radical 2. Once we get one on the hypotenuse we can get r and figure out the points. For the angle of 45 degrees we plot radical two over two, radical two over two. This is where it will lie on the unit circle. This will also stand for 135, 225 and 315 degrees. The only differences are the quadrants the negatives.

3. The 60* Triangle

 

The 60 degree triangle is the same as the 30 degree triangle it has three different sides. In order to derive this triangle from the unit circle is to divide the hypotenuse to get one. This is the same as the 30 degree triangle because you divide by 2x. When you get your final answers you will be able to plot the points. The points for 60 degrees are 1/2, and radical 3/2. These rules also apply for 120,240, 300 degrees.

4.
This activity helps me derive the unit circle because the triangles reflect different points on the unit circle throughout all of the four quadrants. Each of these triangles can be found in all of these quadrants and are all the same the only exception is that there are negatives and positives and they are located in different quadrants too.

5.
The triangle in this activity lies in quadrant one both the x and y values are positive which means that it is in quadrant one.

Inquiry Activity Reflection:
1. The coolest thing I learned from this activity was how you can find the points on the unit circle by using the special triangles.
2. This activity will help me in this unit because it can help me memorize where different points are and where some points lie on different quadrants.
3.Something I have never realized before about the special right triangles and the unit circle are that both these are in relation to each other when the hypotenuse is equal to one