What is the unit circle and why is it important?
Using the unit circle we can figure out the distance between the center and a point on the circle. What’s interesting about this is we can use properties of 45:45:90 degree and 30:60:90 degree triangles. These properties also allow us to also use trigonometry identities to figure stuff out.
From the diagram below that might not be immediately apparent, since there are no triangles “carved” into the circle. Instead we get degree measures: the 90 degree measure is a right angle; the straight line measure is 180 degrees; and entire circle (you might be able to guess) is 360 degrees. Since the entire circumference of a circle is 2π, this length corresponds to 360 degrees. π corresponds to 180 degrees and π/2 corresponds to 90 degrees. By further halfing things, you can find what degree measure corresponds to the length of the minor arc. For instance, half of 90, which is 45 degrees, corresponds to (π/2)/2 or π/4.
Photo by Jim.belk
Notice how π/4 has the coordinate points of √2/2 and √2/2. Notice as well that the radius of the circle is 1. This is very important to know because it allows us “carve” trianlges into the unit circle so that each triangle has a hypotenuse has a length of 1. If we make the hypotenuse at 45 degrees, connecting to point √2/2, √2/2 and then dropping down a straight line so that it forms a right angle with the x-axis, the third leg forming from that point and the origin, we have a 45:45:90 triangle. Since the hypotenuse is equal to 1, the shortest sides are equal to 1/√2, or multiplying √2 times the numerator and the denominator, we get √2/2 as the length of each side. So even if those coordinates weren’t there, you’d be able to figure that out based on the length of the radius.
Next week, I’m going to show you how the same applies to other points on the circle—we can use the lenght of the radius to determine unknown points. But in this case, we will use 30:60:90 properties to figure out the length. In the final post, I will show you can solve trigonometry-based questions using the unit circle.