I/D #1: Unit N Concept 7: Deriving the Unit Circle Using Special Right Triangles

Inquiry Activity

In this activity we learned about the three special right triangles. We had to use online sources to figure out the rules for these triangles. We had to label the measurements of the triangles, assuming that the hypotenuse equaled 1.

1)  30 Degree Triangle

A thirty degree right triangle is called this because it consists of three angles that are 30, 60, and 90. The adjacent side is (x), opposite is (y) and the hypotenuse is (r). The adjacent value is √3x, the opposite is X, and the hypotenuse is 2x. We need to simply the three sides, and we are told that the hypotenuse must equal 1. We now know that the value of the hypotenuse (r) =1. To find the other sides we figure out what X is. X=1/2, this is the value of our opposite side (y). Now we need to find the value of (x), simplify (½) x (√3) = √3/2. We can plot these values as ordered pairs. X=√3/2, Y= ½. This correlates with the ordered pair of 30 degrees in a unit circle, as well as the coterminal angles.

2)   45 degree triangle

In a 45 degree triangle we have two angles with 45 degrees, and a right angle. The value of the sides are… adjacent side is (x) =X, opposite is (y) =X, and the hypotenuse is (r) = X√2. So we know that R = 1, but to find the rest we need to find X. We dived the other sides by X√2 and are left with 1/√2. However we can’t leave a radical in the denominator so after rationalizing we get X=√2/2. This correlates to the ordered pair in a UC because the coordinate for a 45 deg is (√2/2, √2/2).

3)   60 Degree Triangle

 The 60 degree triangle is basically a 30 degree angle but the adjacent side is (x) =X, opposite is (y) = X√3, and the hypotenuse is (r) = 2X. To get the hypotenuse to equal 1 you divide 2x by 2x. You also divide x√3 by 2x, and x/2x. In the end we get R=1, X=1/2, and Y=√3/2.

4)  How Does This Help Us Derive the Unit Circle?

This activity helps us derive the unit circle because we learn the reason why we have the coordinates in a unit circle. Looking at a unit circle, for a 45 degree angle we see the coordinates are (√2/2, √2/2). But, why are these the coordinates? These are the coordinates because this was the answer of what we calculated on a 45 degree triangle, X=√2/2 and Y=√2/2.

5) Quadrants

In this activity the triangle were all in quadrant I, the values change because ther signs become negative and positive. However if the 45 degree triangle was on quadrant II the x value would be negative. If the triangle was on quadrant III both x and y values would be negative. If the triangle was on quadrant IV then just the y value would be negative.

Inquiry Reflection Activity

1. The coolest thing I learned from this activity was the measurements of the sides of the triangles correlate to the coordinates of a unit circle.
2. This activity helped me in this unit because now I know that there is a reason for the coordinate, and if I don’t have a UC then I can just calculate it.
3. Something I never realized about special right triangles and the unit circle is in the unit circle there is many triangles that make up the coordinates.