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let's see if we can give ourselves a little bit of practice converting between radians and degrees and degrees and radians and just as a review let's just remind ourselves a relationship and I always do this before I have to convert between the two if I do one revolution of a circle how many radians is that going to be well we know one revolution of a circle is two pi two pi radians and how many degrees is that if I do one revolution around a circle well we know that that's 360 I can either write it with a little degree symbol right like that or I can write it just like that and this is really enough information for us to think about how to convert between radians and degrees if we want to simplify this a little bit we can divide both sides by two and you could have pi pi radians are equal to 180 degrees or another way to think about it going halfway around a circle and radians is pi radians or you've the the arc that subtends that angle is pi radiuses and that's also 180 degrees and if you want to really think about well how many how many degrees are there per Radian you can divide both sides of this by pi so if you divide both sides of this by pi you get one Radian one Radian it would have to go from singular or from plural to singular one Radian is equal to 180 over PI degrees so all I did is I divided both sides by pi and if you wanted to figure out how many radians are there per degree you can divide both sides by 180 so you'd get PI over 180 radians radians is equal to one degree so this is now I think we are ready to start converting so let's convert let's convert 30 degrees let's convert 30 degrees to radians so let's think about it so I'm going to write it out and actually you might remind this might remind you of kind of unit and analysis that you might do when you first did unit conversion but it also works out here so if I were to write 30 this is how my brain likes to work with it I like to write out the word degrees and then I say well I want to convert to radians so I really want to figure out how many radians are there per degree so let me write this down I want to figure out how many radians how many radians are do we have per degree how many radians do we have per degree and I haven't filled out how many that is but we see just the units will cancel out if we have degrees times radians per degree the degrees will cancel out and I'll be just left with radians if I multiply the number of degrees I have times the number of radians per degree we're going to get radians and hopefully that makes intuitive sense as well and here we just have to think about well if I have if I have think of it this way if I have PI radians PI radians how many degrees is that well that's 180 degrees come straight out of this right over here PI radians for every 180 degrees or PI over 180 radians per degree but then this is going to get us to we're going to get 30 30 times PI over 180 30 times pi PI over 180 which will simplify to 30 over 180 is 1 over 6 so this is equal to PI over 6 extreme let me write the unit's out this is 30 radians which is equal to PI over 6 radians now let's go the other way let's think about if we have PI over 3 radians PI over 3 radians and I want to convert that to degrees so what am I going to get if I convert that to degrees well here we're going to want to figure out how many degrees are there how many degrees are there per Radian and the way one way to think about it is well think about the PI and the 180 for every 180 degrees for every 180 degrees you have PI radians honorary degrees over PI radians these are the essentially the equivalent thing this essentially you're just multiplying this quantity by one but you're changing the units the radians cancel out and then the PI's cancel out and you're left with 180 over three degrees 180 over three is 60 and we could either write out the word degrees or you can write degrees just like that now let's think about 45 degrees so what about 45 45 degrees and I'll write it like that just so you can figure it out is there and figure out what that notation as well how many radians will this be equal to well once again we're going to want to we're going to want to think about how many radians do we have per degree so we're going to multiply this times well we know we have PI radians we know we have PI radians for every 180 degrees for every 180 degrees or we could even write write it this way pi radians for every 180 degrees and here this might be a little less intuitive the degrees cancel out and that's why I'd like to usually write out the word and you're left with 45 PI over 180 radians actually let me write this with the words written out for me that's more intuitive when I'm thinking about it in terms of using the notation so 45 degrees times we have PI radians PI radians for every 180 degrees for every 180 for every 180 degrees so we are left with when you multiply 45 times PI over 180 the degrees have cancelled out and you're just left with radians radians which is equal to what 45 is half of 90 which is half of 180 so this is 1/4 so this is equal to PI over 4 radians let's do let's do one let's do one more over here so let's say that we had negative negative PI over 2 radians what's that going to be in degrees well once again we have to figure out how many degrees are each of these radians we know that there are 80 degrees degrees for every pi radians for every pi radians so we're going to get the radians cancel out the PI's cancel out and so you have negative 180 over 2 this is negative negative 90 degrees or we could write it as negative 90 degrees anyway hopefully you found that helpful and I'll do a couple of more example problems here because the more example for this the better and hopefully become a little bit intuitive