now let's talk about the unit circle so what exactly is the unit circle the unit circle is a circle with a radius of one so if we draw a ray at a 45 degree angle and let's say if we turn it into a triangle the hypotenuse of that triangle will be equal to 1. at a 45 degree angle the x and y portion of the triangle have the same value so therefore the point occurs at this value root 2 comma 2 and the x and y values are the same now you want to know these values because let's say if you wish to find sine of 45 degrees sine is equal to let's say this angle is theta sine theta is equal to the y value of this point in the unit circle cosine theta is equal to the x value of any terminal point on the unit circle so keep in mind for the unit circle the radius is always one and when r is one cosine is equal to x and sine is equal to y so if we want to find the value of sine 45 it's simply equal to the y coordinate of the point that is associated with 45 degrees so this is going to be the square root of 2 divided by 2. and you need to know this because it helps you to evaluate sine and cosine functions based on the unit circle now let's talk about some other common values on a unit circle i'm going to focus mostly on quadrant one because if you know the values in quadrant one you could use that to find everything else there's three common values you need to know 30 45 and 60. at a 30 degree angle x value is root three over two and the y value is one over two at a forty five degree angle we've covered this one already it's the square root of 2 com over 2 comma square root 2 over 2. the x and y values are the same now at 60 the x value is one half the y value is root three over two as you can see some of these values repeat at zero degrees and at ninety at zero it's going to be one comma zero and that ninety is zero comma one at one eighty x is negative y zero and at 270 y is negative but x is zero so x is zero y is negative one so let's say if you want to evaluate cosine of 270 degrees you would choose the x value it would equal zero now let's try some other examples based on the unit circle go ahead and evaluate these uh trig functions find the value of sine of 60 degrees and also find the value of cosine 180 and in addition find the value of sine 30 so sine 60 at 60 degrees look for the y value it's the square root of 3 divided by 2 so that's the value of sine 60 cosine 180 choose the x value that's associated with the 180 angle and so cosine 180 is negative 1 and sine 30 choose the y value sine 30 is one half so if you have access to the unit circle or if you commit it to memory you can easily evaluate any sine or cosine function now let's say if we want to find the value of sine 135 cosine 225 and sine of 315 because we didn't have those angles in a circle that we drew so how can we find these values without actually having to memorize the entire unit circle so remember you only need to know the first quadrant and also the values at the x and y axis if you have that you can figure out everything else so let's start with a 45 degree angle at 45 we said that this correlates to a point that's equal to the square root of 2 over 2 and the x and y values are the same now 45 and 135 are similar an angle of 135 creates the same reference angle as 45 so this is 135 measured from the positive x-axis but it creates a reference angle of 45 with the negative x-axis therefore the x and y-values are going to be very similar at 135 everything is the same except x x is negative in quadrant two but y is positive but the numbers are still the same square root two over two that hasn't changed the only thing that changes is the sign so if we wish to evaluate sine of 135 we need to use the y value so it's equal to positive square root 2 divided by 2. now at 225 it also forms a reference angle of 45 so therefore these values will be the same only the signs will change so at an angle of 225 these are the coordinates it's negative square root 2 divided by 2 comma negative square root 2 divided by 2. in quadrant 3 both x and y are negative so therefore if we wish to evaluate cosine of 225 degrees it's going to be the x value so it's going to be negative square root 2 divided by 2. now let's go over 315 which also have a reference angle of 45. so at 315 x is positive but y is negative in quadrant four and we're looking for the y value for sine three fifteen so therefore it's negative root two over two so that's why if you know the angle in quadrant one you don't have to memorize the values for quadrant two three and four you can figure it out based on the symmetry of the graph and i'll give you some more examples of this let's find the value of sine pi over four i mean pi over three cosine two pi over three sine four pi over three and also cosine five pi over three now if you recall pi is equal to 180 degrees so pi divided by 3 180 over 3 is 60. so pi over 3 corresponds to an angle of 60 degrees now this is the one that we need to know at pi over 3 or 60 degrees you need to know that the point that it corresponds to is this one x is one half but y is the square root of three over two now from this value you could find the other four values so i'm just going to write this in radians pi over three two pi over three is in quadrant two and then we have four pi over three and five pi over three all of these share the same reference angle of 60 or pi over 3. the reference angle is between the terminal side and the x-axis so here they all have the same reference angle now let's write the values at 2 pi over 3 x is negative but y is positive and so understanding this process can help you to memorize the entire unit circle but you really need to know the values in quadrant one after that for quadrants two three and four just change the sign accordingly the x values are negative in quadrants uh two and three that's uh these two quadrants and y is negative and three and four now in quadrant three both x and y are negative so this is gonna be negative one half negative root three over two everything is based on this value in quadrant four x is positive but y is negative so now we can evaluate sine pi over three so pi over three we need to use the y value for sine so this is equal to the square root of three divided by two now cosine two pi over three we need to use the x value at two pi over three so that's negative one half now sine 4 pi over 3 and we need to use the y value so that's negative root 3 divided by 2 and finally for cosine 5 pi over three we need to use the x value which means it's positive one half so now you know how to evaluate trigonometric functions using the unit circle go ahead and find these values sine pi over six cosine five pi over six cosine seven pi over six and sine 11 pi over 6 so pi over 6 that's 180 divided by 6 so that is equal to 30 degrees so feel free to pause the video use what we've talked about to evaluate those functions so let's draw a circle so this is going to be pi over 6 5 pi over 6 which is 5 times 30 that's 150 7 pi over 6 is 210 11 pi over 6 is 3 30. now at pi over six x is equal to root three divided by two but y is one half at five pi over six x will be negative y is positive at seven pi over six both x and y are negative and that eleven pi over six in quadrant four x is positive but y is negative now let's go ahead and evaluate sine pi over six so at pi over six we need to use the y value to evaluate sine so sine pi over six or sine 30 is positive one half now cosine five pi over six we need to use the x value so cosine five pi over six is equal to negative square root 3 divided by 2. cosine is always negative in quadrant 2. and quadrant 3 cosine is negative as well but cosine 7 pi over 6 is going to be the same negative root 3 divided by two and finally sine of eleven pi over six so we need to use the y value and sine is negative in quadrant four so it's negative one half