Focusing on the top right part of a circle we have two special right triangles inscribed: 30-60-90 and 45-45-90. Within the first quadrant of a circle, we have are three main points. They can be found and explained by these two special triangles.
Triangle #1
As we can see this triangle is formed by creating a 30 degree angle. To find our point on the circle, we must know the rules of these special right triangles. Each side of a triangle is labeled with a missing variable. For example, for this special right triangle, the hypotenuse is equivalent to 2n, the shortest side of the triangle is equivalent to n, and the longest side before the hypotenuse equals n radical 3. All we need to do is find the lengths of x and y of the triangle. All we have to do is set the hypotenuse (2n) equal to 1 and we can solve from there on. work is shown below.
Triangle #2
The next special right triangle is created by a 45 degree angle. What we must know is that for this triangle the hypotenuse is equivalent to n radical 2, and the the two other sides are equivalent to n. Again all we must do is set the hypotenuse equal to 1 and solve. From there we can get the lengths of x and y, and eventually our point on the circle.
Triangle #3
For this special right triangle, we can see that it is pretty much the same as the first triangle. Once we do this last one we can see that we have found three coordinates, and these coordinates correspond on a unit circle. This occurs throughout the unit circle except in the other quadrants we must face the signs of the coordinates. We can see that these three points are the same throughout the unit circle and how it relates to it.
