in this video we're going to talk about how to memorize the unit circle now the first thing we need to be familiar with are the angles of the unit circle in radians so here we have 0 degrees and a full circle is two pi half a circle is one pi and half of pi is pi over two so you can think of this as one pi over two pi is the same as two pi over two and this is three pi over two and four pi over two is the same as two pi which will take us back to zero now what we're gonna do at this point is break up the unit circle into eight equal parts now the first one is going to be one pi over four two pi over four reduces to pi over two and then here we're gonna have three pi over four and then four pi over four simplifies to pi this is five pi over four six pi over four reduces to three pi over two and this is seven pi over four and eight pi over four is the same as two pi now the next thing we're going to look at is this angle which is 1 pi over 6 and then 2 pi over 6 reduces to pi over three three pi over six is the same as pi over two next we have four pi over six which reduces to two pi over three and then over here five pi over six we can't reduce that to anything so we're going to write that six pi over six is pi next we have seven pi over six and then eight pi over six reduces to four pi over three so that's a new one we'll have to put that over here nine pi over six is three pi over two and then 10 pi over 6 if you divide 10 and 6 by 2 you get 5 pi over 3 and 11 pi over 6 can't be reduced so we're going to write that here and so that's how you can get the other angles in radians now we need to talk about the values that correspond to these angles so let's focus on the values that are located on the x-axis and on the y-axis let's see if i can fit this one here so on the x-axis to the right x is going to be one for the unit circle the radius is one on the left side x is negative one on the y-axis x is always zero x is zero at the center now for the y values y is zero on the x axis at the top y is positive one at the bottom y is negative one so that wasn't too bad was it now let's focus on quadrant one quadrant one is located on the upper right corner quadrant two is located in the upper left corner quadrant three is located at the bottom left and quadrant four is located in the lower right corner in quadrant one both x and y are positive now let's write the values that corresponds to pi over 3 pi over 4 and pi over 6. so let's go in this direction first let's focus on the x values so we're going to increase it from 0 to 1. so this is going to be 1 over 2 or you can think of it as square root 1 over 2 and then square root 2 over 2 square root three over two and this is equivalent to square root four over two which is one now for the y values we're going to increase it from zero to one in this direction so after zero it's going to be square root one over two and then square root two over two and finally square root three over two and square root four over two is one that's a quick and simple way to find out what these values are on the right side now to find the other values it's going to be a reflection of what you see here across the y axis the only thing that will change is the signs in quadrant one we said that both x and y are positive in quadrant two x is negative y is positive x is negative on the left side y is negative below the x axis so let's go ahead and fill in the values for quadrant two so notice what we see here square root 3 over 2 and one half you want to copy that on the left side so this is going to be the same thing on this side however the x value will be negative that's the only difference so exactly what we see at pi before we're gonna copy that for three pi over four just making the x value negative and what we see for pi over three we're going to copy it for two pi over three but it's going to be negative one half instead of positive one half now in quadrant three both x and y are negative so all of the values with pi over six like pi over six seven pi over six five pi over six eleven pi over six they will all have this value square root 3 over 2 comma 1 half but the signs will vary based on the quadrant it's located in so 7 pi over will have the same values as pi over six however both x and y will be negative five pi over four will have the same values as pi over four but x and y are both negative in quadrant three and four pi over three is going to be similar to pi over three but everything is going to be negative now in quadrant 4 x is positive on the right side but y is negative below the x-axis eleven pi over six is going to be similar to pi over six but only y is negative this time seven pi over four will be similar to pi over four and five pi over three is going to be similar to uh one pi over three but y is negative so hopefully you can see everything i had to squeeze as much info as i can into the small page now you need to know the angles in degrees and you could simply convert the angles from its radian value to its degree value and just know this pi is equal to 180 degrees so to convert an angle measure from radians to degrees just replace pi with 180 so here we have pi over six 180 divided by six is thirty here we have pi over 4 so 180 divided by 4 that's 45 degrees and this is pi over 3 so 180 divided by 3 is 60. this is pi over 2 180 divided by 2 is 90. 2 pi over 3 so 180 divided by 3 is 60 times 2 that's 120 3 pi over 4 180 divided by 4 is 45 times 3 that's 135 and then 5 pi over 6 180 over 6 is 30 times 5 you get 150 and then pi is 180 7 pi over 6 so i think it's pi over 6 times 7 so that's 30 times 7 you get two ten five pi over four it's five times this value so five times forty five is two twenty five four pi over three it's four times this value four times sixty is two forty three pi over two it's three times pi over two or three times ninety so that's two seventy five pi over three it's five times pi over three which is five times sixty and so that's three hundred this is seven times pi over four seven times forty five is three and 11 times pi over 6 11 times 30 is 330 and so that's how you can quickly populate the angle measure in degrees in the unit circle now the last thing that i want to do before i conclude this video is i want to show you how to use the unit circle to evaluate trig functions so i need to make some space because i'm out of space let's say if you want to evaluate sine of pi over three to use the unit circle locate the angle pi over three and if you wish to evaluate sine you need to use the y value so sine of pi over 3 is equal to positive square root 3 over 2. sine of let's say 5 pi over 4 is going to be so first locate 5 pi over 4 and then select the y value so sine 5 pi over 4 is negative square root 2 over 2. now when evaluating a cosine function you need to look at the x value so let's say we wish to evaluate cosine seven pi over six so locate the angle and then use the x value so this is x and this is y so that's going to be negative square root 3 over 2. if you wish to evaluate a tangent function let's say tangent of let's use pi over three tangent is basically y over x it's sine over cosine so take the y value which is square root three over two and then divided by the x value which is one half now if we multiply the top and the bottom by two the twos will cancel and so we're going to get the square root of 3 over 1 which is simply the square root of 3. and so that's how you can evaluate a tangent function using the unit circle so that's all i got for this video if you like it feel free to comment like and subscribe thanks for watching