now let's talk about reference angles and how to find them so let's say if we have an angle of 120 degrees what is the reference angle so let's draw 120 degrees 120 is in quadrant two so this is 120 relative to the positive x-axis the reference angle is the angle between the x-axis and the terminal side or the terminal array and it's always less than 90. it's between 0 and 90. so what is the angle here to find that angle we know it's if this angle is 180 and this is 120 it has to be the difference between the two so it's 180 minus 120 so this angle is 60 and that's the reference angle so let's try another example what is the reference angle for an angle that's 210 relative to positive x-axis so this is 210 and we know the negative x-axis is 180. so the difference is 30. so the reference angle is always between a terminal ray and the x-axis now there are some equations that you can use to easily find the reference angle so anytime you have an angle in quadrant one the reference angle is equal to the angle in quadrant one if you have an angle in quadrant two the reference angle is going to be 180 minus the angle in quadrant two if the angle is in quadrant three the reference angle is going to be the angle in quadrant three minus 180 now if the angle is in quadrant four then it's going to be 360 minus the angle in quadrant four try these two examples find the reference angle for 150 and also 315 so 150 is in quadrant two so this is 150 and to find the reference angle in quadrant two we can use this formula it's going to be 180 minus the angle in quadrant two which is 150 and so the reference angle is 30. so 30 is between the negative x-axis and the terminal rate now on the example on the right for an angle 315 that is in quadrant four to find the reference angle in quadrant 4 it's 360 minus the angle in quadrant 4 which is 315 so that's going to be 45 now if you want to understand it visually a full rotation is 360. and this is 315. so then this missing angle here is the difference between 360 and 315 so therefore that's 45 and that's the reference angle now what if you have a negative angle for example let's say negative 150 or negative 240. how can you find the reference angle well if you don't want to do it graphically the first thing you should do is find the coterminal angle negative 150 plus 360 is equal to 210 now 210 you know it's in quadrant three it's between 180 and 270. so to find the reference angle for an angle in quadrant dream it's that angle minus 180. so the reference angle is going to be 30. now let's analyze it graphically so let's draw the angle at negative 150 so this is negative 90 and this would be negative 150 keep in mind negative angles you need to rotate in a clockwise direction and we can clearly see that this angle has to be 30 because a straight line or straight angle always add up to 180 so you can easily see the answer graphically it's always the angle that's formed if you draw a triangle or between the terminal side and the x-axis now let's try negative 240. so this is negative 90 negative 180 and here's negative 240. it's in quadrant two let's find the coterminal angle let's add 360 to it if we add 360 we get 120. so this angle is also equal to 120. and we know it's in quadrant two so the reference angle is going to be 180 minus the angle in quadrant two that's 120. this works if the angle is positive and between 0 and 360. the formulas that i've given you so the reference angle is 60. and you can clearly see that this angle here is 60 between the x-axis and the blue line now what if we wanted to find the reference angle of an angle in radians what can we do so what is the reference angle of 2 pi 3 it turns out that there's a nice and simple trick to find the reference angle and it only works for common angles in the unit circle like four pi over three five pi over six or seven pi over four if you get some angle that's not on the unit circle like two pi over five this technique doesn't work for it so 2 pi over 3 4 pi over 3 5 pi over 3 negative 4 pi over 3 even seven pi over three all of these they share the same reference angle of pi over three now let's say if we have for example five pi over four or three pi over four or even negative seven pi over four or negative eleven pi over four all of these they share the same reference angle pi over four five pi over six seven pi over six eleven pi over six or even negative seven pi over six or something more than 11 pi over six let's say uh 17 pi over six all of these common angles they share the same reference angle of pi over six so if you see a common angle you can literally see what the reference angle is going to be now what if it's not a common angle for example let's say if we have 3 pi over 5 how can we find a reference angle for this now we need to know what quadrant this angle is located in and if you can't tell what quadrant it is just by looking at the way it is i would recommend converting it to degrees first and do what you did before so let's go ahead and do that let's multiply by 180 divided by pi so we have three times now 180 divided by 5 is 36 so this is 3 times 36 if we multiply that by 3 this is 108 so 108 is in quadrant two and to find the reference angle in quadrant two it's going to be the angle this can be 180 minus the angle in quadrant two so in this case 180 minus 108 so this will give us an angle of 72. now let's convert 72 back into radians so let's multiply by pi over 180 now we know that 180 is 36 times 5 and 72 is 2 times 36 so we can cancel at 36. so 72 is equal to 2 pi over 5. so that is the reference angle now let's try some more examples so let's say if we have an angle that's nine pi divided by eight find the reference angle of this angle so feel free to pause the video and work on it so first let's begin by converting it to degrees so hopefully you have access to a calculator 180 times nine divided by eight as a decimal is about 202 now 202.5 this angle is in quadrantry it's between 180 and 270. so we need to use this formula to find a reference angle it's going to be the angle in quadrant 3 minus 180 so therefore that's going to be 202.5 minus 180 which is about 22.5 degrees so this is the reference angle in degrees but now let's convert it back into radians because we was given an angle in radians to start with so let's take this value and let's multiply it by pi divided by 180 now what i would do is divide it backwards 180 divided by eight i mean divided by 22.5 is eight so you can view 180 as being 22.5 times eight so you can cancel at 22.5 and so the reference angle is simply pi divided by eight let's try one more example but an angle that is still in radians but has a negative value so let's try negative 8 pi divided by 9. so let's multiply by 180 over pi to convert it to degrees now 180 divided by 9 is 20 and 20 times 8 is 160 so this is equal to negative 160 but now let's find the positive coterminal angle so let's add 360 to it 360 minus 160 is positive 200 and positive 200 can be found in quadrant three so now that we know we have an angle in quadrant 3 we can use this formula it's going to be the angle in quadrant 3 minus 180 so in this case that's 200 minus 180 which is 20. so 20 degrees is the reference angle now let's convert 20 back into radians so let's multiply by pi divided by 180 so we can cancel a zero and so we have two pi divided by eighteen and eighteen is nine times two so we can cancel a two so therefore the reference angle turns out to be pi over nine and so that's the answer for this example