What is the best way to memorize the unit circle?
By looking at the Circle notice the points that are the easiest to know are (1,0),(0,1),(-1,0) and (0,-1). By memorizing the degrees start at 0 and add up to 30 once 15 every 2 times and up to 30 again which gives 90 degrees. The same process happens as well for the other degrees. For the other non integer coordinates, start by knowing at 30 degrees is (sqrt3/2,1/2) 45 degrees is (sqrt2/2,sqrt2/2) and 60 degrees is (1/2, sqrt 3/2). Notice that the first coordinates we did were in quadrant 1 which are (+,+) so the next points moving counter clockwiseare in quadrant 2 which are (-,+) So start off by plugging in the values from quadrant 1 using the values that were put in from the similar denominator like 2pi/3 is (-1/2,sqrt3/2) is the one you put in since pi/3 is (1/2,sqrt3/2). The ones after 180 degrees are in quadrant 3 so the points are used in (-,-) form as well as after 270 degrees in quadrant 4 which are (+,-). For the pi's just memorize pi/6,pi/4,pi/3 in quadrant 1 and 2pi/3,3pi/4,5pi/6. Notice for example in 2pi/3 if you add 2 and 3 it gives you 5pi/3 in the same concept in the line that is given to 5pi/3 and vice versa works as well but subtracting 5 and 3 so doing that makes it easier to piut the pi's in the unit circle.
Answer
By looking at the Circle notice the points that are the easiest to know are (1,0),(0,1),(-1,0) and (0,-1). By memorizing the degrees start at 0 and add up to 30 once 15 every 2 times and up to 30 again which gives 90 degrees. The same process happens as well for the other degrees. For the other non integer coordinates, start by knowing at 30 degrees is (sqrt3/2,1/2) 45 degrees is (sqrt2/2,sqrt2/2) and 60 degrees is (1/2, sqrt 3/2). Notice that the first coordinates we did were in quadrant 1 which are (+,+) so the next points moving counter clockwiseare in quadrant 2 which are (-,+) So start off by plugging in the values from quadrant 1 using the values that were put in from the similar denominator like 2pi/3 is (-1/2,sqrt3/2) is the one you put in since pi/3 is (1/2,sqrt3/2). The ones after 180 degrees are in quadrant 3 so the points are used in (-,-) form as well as after 270 degrees in quadrant 4 which are (+,-). For the pi's just memorize pi/6,pi/4,pi/3 in quadrant 1 and 2pi/3,3pi/4,5pi/6. Notice for example in 2pi/3 if you add 2 and 3 it gives you 5pi/3 in the same concept in the line that is given to 5pi/3 and vice versa works as well but subtracting 5 and 3 so doing that makes it easier to piut the pi's in the unit circle