I/D #1: Unit N Concept 7: Knowing the degrees, radians, and ordered pairs of a Unit Circle

   The special right triangle (SRT) and the unit circle relate to each other in a way that the special right triangle of 30, 60, 90 and 45, 45, 90 can be found in the first quadrant of the unit circle. Moreover, it reveals the ordered pairs on the other three quadrants, depending on its sign (positive or negative). Special right triangles have a hypotenuse of 1 and the unit circle has a radius of 1. Thus, special right triangles can be located in all of the quadrants on the unit circle.

INQUIRY ACTIVITY SUMMARY

 1. 30 degree triangle

  This is the 30 degree special triangle. Across from the 30 degree angle is x, across from the 60 degree angle (not labeled) is x radical 3, and across from the 90 degree angle is 2x. The hypotenuse of this triangle is one.

  This pictures demonstrates a the sides and ordered pairs of the 30 degree special triangle. Since the the hypotenuse or the radius is equal to one, x (across from the 30 degree) has to be 1/2 and x radical 3 (across from 60 degree angle) has to be radical 3 over 2. The work is shown beneath the triangle. You can simply find the ordered pairs by looking at its points. This triangle is labeled at quadrant one on the graph. 

2. 45 degree triangle 

   This is a 45 degree special right triangle. Since two out of the three angles are 45 degrees, the letter x is the same. Across from the 45 degree is x and across from the 90 degree is x radical 2.

   This pictures reveals the ordered pairs and sides of the 45 degree special right triangle. Since the hypotenuse or radius is one, the letter x, across from the two 45 degrees, will be radical 2 over 2. The work is shown below the triangle. The triangle in this picture is located in quadrant one on the graph. Ordered pairs are plotted according to its location.

3. 60 degree triangle

   This is a 60 degree special right triangle. Across from 30 degrees (not labeled) is x, across from 60 degrees is x radical 3, and across from 90 degrees is 2x. This triangle is similar to the 30 degree special right triangle. However, you will noticed that the ordered pairs and sides will be switched.

   This picture shows the ordered pairs and sides of the 60 degree special triangle.  The hypotenuse/radius is one, which is similar to the last two special right triangles. The sides of this triangle is consisted of radical 2 over 2, 1/2, and 1. Work is shown below the triangle. Again, this triangle is labeled on the first quadrant. Ordered pairs are plotted according to its side and point.

4. This activity helps me derive the unit circle because it gives me the degrees, points, and ordered pairs that are located at quadrant one. Thus, with this knowledge, I can apply it to the other three quadrants on the graph. I can use the idea of reference angles to know what degrees  and ordered pairs (depending on the sign) will be on the unit circle for each quadrant. The special right triangle is part of the unit circle in a way that it depicts all the pieces that a unit circle has.

5. The triangles in this activity are located in quadrant one. Quadrant one is used to find all the other parts in the other quadrants. The "magic 5" relates to the other quadrants in terms of degrees, radians, and ordered pairs. The "magic 5" includes 0, 30, 45, 60, and 90 degrees. Knowing the first quadrant of the unit circle is very significant. Nevertheless, this is where the term reference angles and coterminal angles come in need. Coterminal angles are angles that have the same terminal side. Reference angles are positive, acute angles (from degree to x-axis).

http://01.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_34.gif

  This pictures shows the 30 degree special right triangle in quadrant II, III, and IV. All of these three quadrants are similar to the quadrant I. The difference in these quadrants is the ordered pairs because of the signs depending where it's located. Each quadrant has the same reference angle, which is 30 degrees.

 

http://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/ebaa19ac-ff8b-43a6-a793-a00d9ac15e86.png

   This picture shows the 40 degree right triangle in all of the quadrants on the unit circle. The only difference in each quadrant is the sides/ordered pairs because they can either be a positive or negative depending where it is plotted. Reference angles are still the same.

http://01.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_21.gif

   This pictures shows the 60 degree right triangle on the other quadrants (besides first quadrant). The sides are the same such as 1/2 still lies on the x-axis at quadrant III and IV. The difference, similar to the other special right triangles, is the sign on the ordered pairs. The values, otherwise, are still the same according to the first quadrant.

INQUIRY ACTIVITY REFLECTION

1. The coolest thing I learned from this activity was knowing how to relate special right triangles to unit circles.
2.  This activity will help me in this unit because I can now see the pattern and find a way to memorize all the degrees, radians, and ordered pairs in a unit circle.
3. Something I never realized before about special right triangles and the unit circle is how they are all connected to each other, in terms of the radius and the quadrants and all the other pieces that make up a unit circle.

References
http://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/ebaa19ac-ff8b-43a6-a793-a00d9ac15e86.png
http://01.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_34.gif
http://01.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_21.gif