Special Right Triangles and Unit Circle

The rules that special right triangles have can be related and used in a unit circle because a circle also has 30°, 45°,and 60°. The hypotenuse(r) of the triangle or radius of the circle is always equal to 1. So for a 45,45,90 triangle, we already know r is 1 and x/y is equal to 1/2r√2. So x/y is equal to √2/2. On the picture on the left, we see that the coordinates of a certain point on the circle is (x,y) and if we plug in what we just found out, the coordinates for a 45° is (√2/2, √2/2). To find the side lengths of a 30,60,90 triangle we must know that r will always equal 1, y equals 1/2r and x is equal to 1/2r√3. After solving for everything, x turns out to equal √3/2 and y equals 1/2. By filling the coordinate, we find out that the coordinates for a 30° is (√3/2,1/2). To find the coordinate of a 60°, just switch the ordered pair of the 30° and that will be (1/2, √3/2). So we just found 3 out of 5 of our MAGIC 5 and to find the last two, refer to the picture above. The last two coordinates needed to find is the one on the x-axis and the other on the y-axis: the two points that say (r,0) or (o,r). As you can already tell, just plug in the value for r which is always 1 and you have just found your MAGIC 5 and how special right triangles work so well with unit circles.