I/D #1: Unit N Concept 7: How do SRTs and the UC relate?

Inquiry Activity Summary:      

-In the following activity you will learn the rules of Special Right Triangles and how the rules and points that you learn from the triangles, can be tied back to the Unit Circle. The special triangles are considered "special" because there sides and hypotenuse is what the Unit Circle is made out of. The three special triangles are, 30 degrees, 45 degrees, and 60 degrees.

https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgI-XejwVVlrhIrft3u1Srbt4a6MEKSzd0YbGD0oIHHAyeuDBqjUqsQgk2imdYjoRadTfPkAQ6DWKjW9-twnJGEmhQjHbtkVIvaHv2k3cwttL10uNKbr4YPkmljMwH-pOWlS3lz9mt46YVv/s1600/Untitled.png

1. The rules for a 30 degree triangle( focus on the picture to the left) are the following:

- the hypotenuse(opposite of 90 degrees) is 2x because

http://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/f8c8bda9-7e7d-47da-b2b6-a36cc369bf07.png

it is 2 times the short side.

- the shortest of the triangle has the value of x.

-the longer side (opposite of the hypotenuse) has the value of radical 3.

 

2. The rules for a 60 degree triangle (focus on the picture above on the right) are the following

*the same as the 3o degree triangle just flipped

- the hypotenuse(opposite of 90 degrees) is 2x because it is 2 times the short side.

-the shortest of the triangle has the value of x.

- the longer side (opposite of the hypotenuse) has the value of radical 3.

 

3. The rules for a 45 degree triangle are the following: 

http://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/8b5cdca6-5a42-4ff4-b3df-cabf3e95914e.png

  - the hypotenuse has length x times radical 2

- the sides that make up the 90 degree angle are both the length of x.   

4. This activity made me realize that the special triangles and the unit circle have so many things in common. If you refer back to the picture above, in the red circle there are ordered pairs that reflect back onto the unit circle. The picture below does a great job demonstrating how the unit circle is basically all special triangles. The 03, 45, and 60 degrees on the chart show the order pairs in the unit circle. Really take a look of the unit circle and look at how there is a similarity between each of the quadrants and the logic behind the ordered pairs is the rules that are listed above.

http://www.pccmathuyekawa.com/classes-taught/math_7ab/unit%20circle.jpg

5. As the picture that is show above, all the triangles would apply to all four quadrants. If you flip the triangles in the other three quadrants you would notice a pattern in them. Check out the flowing picture and see what you notice. 

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

This picture demonstrates how a 30 degree triangle would look in all the quadrants. Notice that the order pairs do not change, the only things at do are the positive an negative signs that should be easy to understand since in a graph, the second quadrant has a negative x-value but a positive y-value. For the third quadrant both the x and y-value are  negative. Then, in the fourth quadrant, the x-value is positive and the y-value is negative but still the same ordered pairs. See the following example for 60 degree visual.

http://01.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_13.gif

  This is a 60 degree  graph and it will be the same concept as the 45 degree one.

https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgAeFAQcOzN5WJ1oGL3TBnwK0Qnavf3Jec3LauiNNMLy3Xg8gGR1zEL_f6TSjqpkFO4Y67K-WljPwjTP_bLbfLKGDrCQXKZBuppK7uF87kM0ZHlBWnpmWuhIyqfXsVqWY-8naWmCk5fmSm6/s1600/123.png

 This is a 30 degree one and make sure you do not confuse it with the 60 degree one. Other than that same thing applies to this as the 45 degree example.

 

 

 

 

 

 

Inquiry Activity Reflection:

1.  “The coolest thing I learned from this activity was" the unit circle is basically made up of many special right triangles.

2. “This activity will help me in this unit because…” it really made me realize new tricks and patterns that lie within the unit circle all because of triangles. Knowing the ordered pairs for the triangles is easy it is just knowing whether it is negative depending on the quadrant that it falls on.

3. “Something I never realized before about special right triangles and the unit circle is…” I did not realize that the unit circle is special triangles that you only need to know the first quadrant in order to find the rest of them on the remaining quadrants.