Radian and degree conversion practice A little practice converting between radians and degrees and vice versa
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- Lets 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 lets just remind ourselves the 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 (2)(pi) radians
- and how many degrees is that if I do one revolution around a circle?
- Well we know that is 360 degrees.....
- I can either write it with the little degree symbol 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 2.
- And you can have (pi) radians are equal to 180 degrees
- or another way to think about it
- going halfway around a circle in radians is (pi) radians.
- The arc that subtends that angle is (pi) radiusssses
- and that's also 180 degrees.
- And if you wanna really think about, well 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 would get one radian
- it would have to go from plural to singular
- one radian is equal to 180/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 would get pi/180 radians
- is equal to 1 degree.
- So this is....now I think we are ready to start converting.
- So lets convert 30 degrees....to radians.
- So lets think about it.
- I'm going to write it out and actually this might remind you of unit analysis
- that you might do when you first did unit conversion
- but it also works out here.
- So if I were to write 30 degrees
- and this is how my brain likes to work with it
- I like to write out the word degrees.
- Well I wanna convert to radians so I really want to figure out
- how many radians are there per degree?
- So let me write this down. I wanna figure out how many radians do we have per degree?
- And I haven't filled out how many that is
- but we see the units will cancel out
- if we have degrees times radians/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 gonna get radians.
- And hopefully that makes intuitive sense as well.
- And here we just have to think about
- well if I have...think of it this way, if I have
- pi radians how many degrees is that?
- Well that's 180 degrees, comes straight out of this right over here
- pi radians for every 180 degrees or pi/180 radians/degree.
- This is going to get us to...we're going to get 30 times pi/180
- 30 times pi/180
- which will simplify to
- 30/180 is 1/6
- so this is equal to pi/6
- let me write the units out
- this is 30 radians which is equal to pi/6 radians.
- Now lets go the other way
- lets think about if we have pi/3 radians
- and I wanna convert that to degrees.
- So what am I going to get if I convert that to degrees?
- Well here we're gonna want to figure out
- how many degrees are there per radian?
- One way to think about it is
- think about the pi and the 180
- for every 180 degrees you have pi radians.
- 180 degrees/pi radians
- these are essentially the equivalent thing
- essentially you're just multiplying this quantity by 1
- but you're changing the units
- the radians cancel out and then the pi's cancel out
- and you're left with 180/3 degrees
- 180/3 is 60 and we can either write out the word degrees or you can write degrees
- just like that.
- Now lets think about 45 degrees.
- So what about 45 degrees?
- And I'll write it like that just so you can figure it out with that notation as well.
- How many radians will this be equal to?
- Well once again we're gonna 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
- for every 180 degrees
- or we can even 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 usually like to
- write out the word
- and you're left with 45 pi/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 for every 180 degrees.
- So we are left with when you multiply
- 45 times pi over 180
- the degrees have canceled out and you're just left with
- radians.
- Which is equal to what?
- 45 is half of 90 which is half of 180 so this is 1/4
- this is equal to pi over 4 radians.
- Lets do 1 more over here.
- So lets say that we had negative pi/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 180 degrees
- for every pi radians
- so we're gonna get
- the radians cancel out
- the pi's cancel out
- and so you have
- -180/2
- this is -90 degrees
- or we can write it as -90 degrees.
- Anyway hopefully you found that helpful
- and I'll do a couple more example problems here
- because the more example for this
- the better and hopefully it will become
- a little bit intuitive
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