Unit Circle

The unit circle is important to trigonometry, and is usually measured in radians instead of degrees. If you don't know what radians are, that will be explained shortly. A unit circle is a circle placed on a Cartesian grid (or coordinate plane) and has a radius of 1. This is important; this means that the circle has a circumference of 2π, and this is where radians come in. The unit circle is measured in radians, and 2π is the measure of the whole circle, since it is the circumference. Every other measurement is a fraction of this. For example, half the unit circle is π, a fourth is π/2, and 3/4s is 3π/2. If you're working with degrees and want to convert to radians, there is a very simple way. Just multiply the degree measure by π/180. For example, if you want the radian measure for 240, just multiply 240 * π/180, which gets 240π/180, which simplifies to 4π/3. Likewise, to convert radians to degrees, you just go the other way around. Using 4π/3, multiple by 180/π, which is 720π/3π, which simplifies to 240. Works fine, doesn't it? Every measure on the unit circle has an exact point. For example, 0 (which is the same as 2π on the unit circle), is point (1,0). π/2 is (0,1), π is (-1, 0), and 3π/2 is (0, -1). Those are the easy ones, but how do you find the ones in the middle? For example, π/4 (which is right in between 0 and π/2) is (√2/2, √2/2). This is found by extending a line down from π/4 point. It creates a 45-45-90 triangle.



As the image shows, the length of both legs are 1, because the length of the radius of the unit circle is one (although the picture doesn't exactly represent that, but bear with me), and the hypotenuse is √2 because of the 45-45-90 rule. Now, to get the points of the π/4 measure, take the cosine and sine of the 45 degree angle next to the origin (although it really doesn't matter). Recall that sine equals the y value and cosine equals the x value. So, since both legs are 1, both values are 1/√2, which rationalizes to √2/2, and that's how π/4 has the coordinate √2/2. As for the measures next to it, π/6 and π/3, a 30-60-90 triangle is needed, as shown. To get the points for x and y, take cosine and sin again (respectively), and there you go.



All other points on the unit circle can be derived by doing reflections over the x and y axis, and as you go around the unit circle counterclockwise, the radian measures get closer to 2π and the degree measure moves closer to 360. When graphing sine, cosine, tangent, cosecant, secant, cotangent, arcsine, arccosine, and arctangent (that's a lot of graphing), a unit circle is necessary (or a calculator), that is, unless you have it memorized.