I/D #1: Unit N Concept 7: Deriving the Unit Circle Activity

Inquiry Activity Summary:

In this activity, we dealt with Special right triangles: 30,60, 90  45,45,90  and 60, 30, 90:

http://ss.worldindustrialreporter.com/Library/ui/Trigonometry/Technical%20Shop%20Mathematics/Special%20Triangles%20and%20Unit%20Circle/1/default.aspx

The triangle's sides are labeled according to the Special Triangles Rule. The 30, 60, 90 triangle can also be known as: hypotenuse (2x) or (2 as labeled in the picture) opposite side to 30 degree angle as (x) or (1 as labeled) adjacent side as (x radical 3) or (radical 3 as labeled). The 45,45, 90 degree triangle has two sides that have the same value because there are two angles (45) that are the same. The 45,45,90 degree triangle can also be seen as: hypotenuse (x radical 2) or (radical 2) and the two 45 degree corresponding sides will both have the value of the side being (x) or (1).

1) 30 Degree:

The first triangle in the activity dealt with the 30 degree triangle. The first step was to get the hypotenuse to equal 1; the only way to make this possible was to divide ALL sides of the triangle by 2 and then simplify. This gave us the value of each side: HYPOTENUSE=1 ; OPPOSITE SIDE (to 30 degrees)= 1/2 ; ADJACENT SIDE (to 30 degrees)= radical 3/2. Next, I labeled the hypotenuse (r), the horizontal value (x), and the vertical value (y). The next step was to draw the triangle on a coordinate plane. The triangle must lie in quadrant I which means that the origin of the triangle is angle 30, making its ordered pair value (0,0). When we move over the the 90 degree angle, we move a distance of (radical 3/2, 0), then we move up to our 60 degree angle and x distance of radical 3/2 and up distance(vertical) of 1/2 giving us the ordered pair (radical 3/2, 1/2) *notice that these ordered pairs correspond to the side value after we simplified all sides to make the hypotenuse equal to one*

2) 45 Degree

In a 45 degree triangle, the hypotenuse is radical 2 (r), horizontal side is x (x), and the vertical side is x (y). The value of the hypotenuse has to equal to 1 so I had to divide everything by radical 2. This made r=1, x=radical 2/2, y=radical 2/2. Next i drew the coordinate plane for the 45 degree triangle and its origin (45 degrees) was (0,0), 90 degree was (radical 2/2,0), and the last side point was (radical 2/2, radical 2/2). Any other angle with a reference angle of 60, will have the same values throughout the unit circle(signs vary depending on quadrant).

3) 60 Degree

Last but not least, we have the 60 degree triangle. The hypotenuse(r) is 2, opposite (to 60 degrees) is radical 3, and adjacent (to 60 degrees) is x. Since the hypotenuse has to equal 1, I divided everything by 2, this made r=1, (y/ vertical)= radical 3/2, and (x/horizontal)= 1/2. After drawing the triangle on a coordinate plane, it is in Quadrant I. The ordered pairs for the triangle are: (0,0), (1/2, 0), and (1/2, radical 3/2). In ANY angle with a reference angle of 60, the values will be the same.

4)This activity helps us derive the unit circle because once we find the values for the 30, 60, and 45 degree triangles in the first quadrant, we can figure out the rest of the unit circle.

5)The triangle drawn in this activity lies in Quadrant I. This means that ALL values are positive. The values change depending on the restrictions of each quadrant. As seen above, in Quadrant II the x value turns negative in all ordered pairs. In Quadrant III the x AND y value are negative for all ordered pairs. In Quadrant IV, the y value is negative in all ordered pairs.

INQUIRY ACTIVITY REFLECTION:
1. The coolest thing I learned from this activity was that once you figure out all the parts to triangles with 60, 30, and 45 degrees, they basically set you up for the rest of the unit circle.
2.This activity will help me in the unit because it will help me memorize the ordered points and the reference angles to which they correspond to.
3.Something I never realized before about special right triangles and the unit circle is how they existed within each other and how they can help you with any angle as long as you find the reference angle.