The first time you look at this fat..hodgepodge of unrelenting chaos, all you can think is “My god, numbers EVERYWHERE, I'm supposed to remember THIS?”
Well, if you're using Trigonometry, it helps. It really does, please believe me.
Today I want to explain something about the Unit Circle. For all you types who have seizures at the thought of doing high school math, it may not be your cup of tea, but for even a small portion of us, it will become invaluable and maybe even a liiiiiittle interesting. When you want to know how tall something is for instance... or how many stones you will have to buy for that brand new ^%^%ing path from the house to the gate that you have to put in yourself because the estimates from the "pros" would have put your retirement at 103.
I will explain only one part of the Unit Circle. It's called the First Quadrant. I will restrict my explanation to only the angles, not the values that they are associated with.
Plain and simple, the Unit Circle has degrees and radians that are the angles you see inside a triangle. Each quarter of the circle is called a Quadrant. In each quadrant there are points that have both a degree, and a radian. Don't be intimidated by the name "radian" it just means that it has a pi symbol.
The circumference of a circle is 2 times pi, times r. The Unit circle always has a radius (from the middle of the circle to anywhere on the outer edge) of 1. This means the circumference of the whole Unit Circle is 2pi. The very first point we put on the circle is 0. It always starts at the right of the circle, and is the beginning of Quadrant I. The end of Quadrant one is the 90 degree angle right at the top of the circle.
Trigonometry tells us the story of the sides and the angles of triangles. So to get the points of Quadrant 1, we look at two "special triangles." The first is called a 45-45-90 triangle
and the second is a 30-60-90 triangle.
These are both right triangles in which we place one 90 degree angle, and that makes the points that describe the angles on the Unit circle. One 30 degree point right after 0, one 45 degree point in the middle, one 60 degree point, and of course 90 degrees at the end of our first quadrant.
Now we have all the degrees we need. So what else do we need? Well, degrees, believe it or not, are not very accurate. Yes yes, I hear you yelling now "But ships and planes and stuff use them!" Yes, and they do a very fine job. But can you imagine trying to measure the tip of your pencil with a yardstick? This is what it is like for those who are using Trigonometry with sciences in which they need smaller numbers, finer, more exacting detail. Degrees sometimes just won't cut the mustard. So we have Radians. Radians is like using a fine tip sharpie to paint that moustache on Amerigo Vespucci's butt in your textbook, instead of that massive, thick nubbed pen which makes it look like he's...well..incontinent. con..ti..nent....get it? oh never mind.
At any rate, here we are. We know all our degrees, how do we get our radians? Well. We know that the entire circumference of a circle is 2pi. Try cutting that in half. Go ahead. Good!! it's pi. This is half a circle.
Now, to get to the end of the first quadrant, or a quarter of the Unit circle cut pi in half.. ah yes, pi/2. This equals 90 degrees.
Well, we have two points already. We nearly have the whole thing solved. This may become...a little...tricky.
Cut pi in half again. Don't be shy. Just divide pi/2 by 2. Remember that when you divide a fraction by a number, you can get the answer by multiplying that fraction by the reciprocal of the the other number!
Did you get pi/4? So did I!! Guess what, that equals 45 degrees.
So what now? we have two angles left. How do we get 30 degrees? Well there are 3 points separating 90 degrees or pi/2 from 30 degrees just as there are two points separating 90 degrees from 45 degrees. So lets divide pi/2 by 3.
pi/2 divided by 3 = pi/6. Good job!
There is a difference of 30 degrees between 0 and 30, and 15 degrees between 30 to 45, then 15 degrees between 45 to 60. So if we add 30 degrees and 30 degrees, we come up with 60 degrees. Since we already know that 30 degrees equals pi/6, we should be able to add pi/6 to itself, and we will get....pi/3. When you go "but wait...I got 2pi/6!" Remember that 2pi/6 is a fraction and we must reduce. 2 goes into 6 a whole 3 times giving us pi/3.
Now you know the radians and the degrees for the first quadrant. By remembering the changes in the first quadrant of the unit circle, you can figure out every quadrant there after by adding multiples of pi/6, pi/4, and pi/3. But just in case you get stuck with radians, here is a tip. After you figure out all your degrees for each quadrant, you can change those degrees to radians, by simply multiplying the degree by pi/180. for instance, Multiply 30 degrees times pi/180 degrees. The degrees cancel, 60 goes into 160 six times, and you are left with...
You can do this for any degree on the Unit circle and it will work. But do try to add multiples of pi/3, pi/4, and pi/6. Learn to see the patterns that occur between the fractions, and you will also reinforce your basic math skills.
By the way ... you won't buy enough stone for the path to the gate. You'll go back twice before you do, then it will rain, making work impossible for two weeks. You'll stare broodily out the window wondering how the gods could hate you so much. You should probably quit your job and become a writer with all those violent thoughts. Make a million bucks.
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