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The Unit Circle

A brief introduction to memorizing the complete unit circle, including measures of radians, intercepts and degrees in both 8ths and 12ths.

Elizabeth Paulukonis

on 26 September 2011

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Transcript of The Unit Circle

Memorizing the Unit Circle Coordinates Radians Degrees The Unit Circle in 8ths The Unit Circle in 12ths Think in Terms of 45 Think in Terms of 30 From zero, every angle is either 30+ or 45+ Unit Circle in 12ths Unit Circle in 8ths sqr rt(2)/2, sqr rt(2)/2 Remember that for each quarter
of the unit circle, the x and y coordinates remain the same, but whether or not they are positive or negetive depends on the section they're in.
1/2, sqr rt(3)/2 It's easiest to think in terms of two in this case - for each quarter of the unit circle, the x and y coordinates switch. For example, in the I quadrant, at 30 degrees, the x coordinate is sqr rt(3)/2, and at 60 degrees, it's 1/2, with the y coordinate at each seperate point being the opposite coordinate. The x and y coordinates are also negative and positive according to what quadrant they're in - the tricky part is remembering which comes first. For me, it's easiest to look across at the opposite degrees - for example, at 60 degrees, I know that 1/2 comes first, so at 240 degrees, the 1/2 will come first, AND be negative (along with sqr rt(3)/2). Likewise, at 120 degrees, 1/2 is the x-coordiante, so at 300 degrees, the 1/2 will be the x-coordinate as well, with sqr rt(3)/2 being the negative y-coordinate.

Memorizing radians is the most difficult
part of the unit circle - I find it's easiest to
start with the "essentials." 0 pi
pi/2 3pi/2 2pi All of these points
define the four sections of
the unit circle - I through IV.
2pi is the complete rotation -
it ends where 0 started. Radians in 8ths The best way for me to remember
the four radians in between the "essentials"
is to remember that first of all, every denominator is 4.
From there, it's a matter of remembering what's on top. Pi is in the first quadrant, and in the second, we out 3 on top, too. The last two are easy - just add 2 to what you previously had, and voila - you're done. The "Essentials" Radians in 12ths Let's start off by remembering, again, our essentials. They're the same as they were for 12ths. All we need to do in this case is remember the 8 radiansin the four quadrants. Let's think about the denominator first. 3 6 & Denominators To remember which radian comes first, remember that 6 is always closer to the x-axis, and 3 is always closer to the y-axis. Now to remember what goes on top is the hard part - let's start by thinking in terms of quadrants. Pi goes over the denominators in quadrant I - easy enough. What about quadrant II? It's just easiest for me to remember - 2pi over 3, and 5pi over 6. The III quadrant? Same deal - remember that 7pi is over 6, and 4pi is over 3. IV? 11pi over 6 and 5pi over 3. The Unit Circle Let's see if you remember by Monday...
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