hey Jonathan here at Colfax math this channel is the applications of math in
all things math today I'm going to talk about Radian measure comparing it to
degrees this is a really big idea in both math and science and it's really
cool as well and there's a there's a few pieces to it and let me run through
those right now so let me start with degrees degrees is a unit measure you
can measure an angle and degrees it's usually a central angle off of the
circle the notation for the measure of an angle in degrees is that little
circle afterwards and a full rotation is 360 degrees so a half circle is 180
degrees a quarter 90 and the 3/4 circle 270 degrees I could cut those in half to
get 45 135 225 315 I could split it into eights I could split it into six or 12
and those are degree measure the Greek letter for measure of an angle is Theta
and there's no way to tell if it's radiating degrees except for by the
units that are labeled on there so if you see this when there's no notation of
yet it has to be radians when you see this it has to be degrees so radians are
actually unitless so when you see theta equals 20 and
there's no notation to units it is implied that there are no units and it's
radians okay so that's the first way to figure out the difference between a
degree and a Radian the idea that radians are actually unitless is what
makes them so useful because then you could take this math and apply it to any
real-world application problem okay so what is a Radian a Radian is a measure
of radiuses so if this is my radius here I call this radius
our and we were to measure an arc length right here when that arc length is the
exact same length as the radius I could also call that R then I can measure this
arc length in terms of radiuses and it would be one radius long let's say this
R comes with eat even longer like this like I say I don't know this is two
units long my radius is one unit say I want to figure out what this arc length
is in terms of radiuses that our client would be equal a 2/1 or two radiuses so
I can talk about that arc length being two radiuses long where that arc length
could be three radius is long or four radiuses long so Radian is a way to
measure arc length in terms of the radius so let's let's go back a step and
explain this another way I know the circumference of a circle is equal to
two PI R or diameter times pi right so now if my circumference is an integer
say and I have two pi r r would also be in inches my radius would be in inches I
could divide both sides of this equation by two are in inches two are in inches
right then I could see that my T's would cancel my radiuses would cancel my
inches would cancel and over here on this side my inches cancel I could see
that pi is a ratio of circumference to 2r and there are no units attached to pi
because my units have cancelled out here and that's why radians our unit
okay it's a big idea and the whole point of the radians being unitless is that
you can use it in a lot of different scenarios when you know if you're
measuring an electrical wave it's not measured in degrees it might be in time
and time isn't degrees so that's why you use units of radians so that you could
attach seconds or whatever you want to it so remember pi is not a variable it's
a number it's a ratio or circumference to diameter or to radius my units cancel
and pi is unitless all right let's go over a few problems first so if all the
way around the circle is 360 degrees in the unit circle and my radius is 1 the
circumference would be 2 pi our radius is 1 so all the way around would be 2 pi
halfway around would be pi I'd want to cut that in half so if this is 2 pi pi
this would be 1/2 pi 1 pi 1 and 1/2 pi 3 PI over 2 so these are exact value
measures in radians and a black these are exact value in degrees but they're
equivalent the other thing about radians is it could be used to measure an arc
length or an angle contained within so that's kind of a cool part of it as well
okay let me turn the page here so let's just do some conversions from degrees to
radians so if I have 30 degrees and I want it to be into radians I'm only
writing radians as a placeholder I don't actually have to write radians because
if nothing's written its radians so it's a unit conversion well if I know
right here that 360 and 2pi are equivalents I know that 2pi and 360 are
the exact same thing right so the same things that this has the same way to
this so this is equal to 1 or I could reduce this because this is equal 2
times pi divided by 360 is 2 times 180 so the twos cancel and give me pi over
180 so this is my conversion unit to go from degrees to radians so to go from
degrees to radians I use PI over 180 and to go from radians to degrees I use the
reciprocal 180 over PI ok that's my conversion unit so 530 degrees
I want to go to radians I have to be used to radians I'm going to multiply by
PI over 180 these things have the exact same value the PI in the 180 so it's
equal to one I'm multiplying by one my degrees cancel and that's gonna give me
30 PI over 180 but then I could reduce that as well because 30 goes into 31
times 30 goes into 180 6 times so it's equal to PI over 6 radians so 30 degrees
is PI over 6 radians Steve another one here let's say I have 45 degrees and I
want to convert that into radians I multiply by PI over 180 degrees my
degrees cancel 45 goes into 180 4 times so this gives me PI over 4 so that's my
answer it's in radians I don't have to write radians again
presents unitless I just write it as a placeholder if it's degrees I always
have to have that circle okay so now let's go the opposite way
let's go from radians to degrees so from radians to degrees I multiply by a
conversion factor of 180 degrees over pi so if I have say PI over four and I want
to convert that into radians I multiply it by 180 over pi
there's my radians that's a hundred and eighty degrees I multiply by that
conversion factor my PI's cancel four goes into here one time four goes into
here forty five times so it gives me 45 degrees okay let's say if I just had a
Radian measure of pi I multiply by the same conversion factor 180 degrees over
pi my PI's cancel and it gives me a hundred and eighty degrees I have to
write that unit down every time because now I'm in degrees okay so that's how I
convert from radians to degrees actually why don't you do one really quick here
let's say I have 3 PI over 4 do you want to pause the video and do that one 3 PI
over 4 convert that into degrees okay so what you do or hopefully what you did is
you multiply it by 180 degrees over pi
my PI's will cancel 4 will go in a 180 45 times 45 times 3 right 45 times 3 is
135 degrees all right wanting to try go in the
opposite way so let's say you have a hundred and twenty degrees go ahead and
pause the video and convert that into radians okay convert that to radians I
multiply not by 180 over pi but I want Radian so it's by PI over 180 here my
degrees cancel and then I want to reduce this fraction 120 PI over 180 60 will go
into here twice 60 will go in here three times by giving PI over three and no I'm
sorry two pi two pi over three and that's the equivalent of 120 degrees
again I don't need to write radians on here for this implied it's in radians
because rate into a unitless if I wrote radians on here it's just kind of as a
placeholder but it's kind of redundant okay so that's how you convert radians
to degrees and back again my last idea I want to put out on this here is the
difference between the exact value and approximate value okay so exact value
means there's no rounding errors at all so the number pi is an exact value
versus a decimal approximation it's an approximate value would be 3.14 so
that's the decimal approximation of the number pi another example would be say
the square root of 2 not sure what that is but I think it's 1.4 the decimal
approximation of that is 1.4 1/4 or an exact value if you say one-third and
fractional form the decimal approximation
would be 0.33 repeating so exact values are always better answers position Oh
rounding error there are no significant digit issues with it
so whenever you could leave it is an exact value you want to so the exact
value is without a calculator approximate value is with a calculator
and how all those kind of plays into exact approximate values plays into
degrees and radians is let me draw my unit circle here again remember all the
way around 360 halfway around 180 9217 or the Radian equivalence this would be
pi this would be PI over 2 this would be 3 PI over 2 and this would be 2 pi so
this is degrees oops I forgot very importantly my degree notation but I
have to say that these things are degrees so in black here these are
degrees these are radians these are exact values
of radians so I could call it 2 pi or I could still be in radians and call it
3.14 times 2 6.28 that's still a Radian measure it's just
an approximate Radian measure versus the exact value Radian measure so it could
be 360 degrees that's a different type of units
that's degrees 2 pi is a Radian 6.28 is a Radian and the difference is exact
verse approximate I could call this pi over 2 in Radian or 1.57 radians and
that's my approximate value the way I got 1.57 I just use my calculator and
took PI 3 point one for divided it by two so usually in
most trig courses you should know the 30-60-90 triangles and the reference
angles and also the 45 and maybe I'll do another video on that just touch on it
here I'm gonna go in degrees first half of 90 is 45 that gives me you know it's
an isosceles right triangle one Wow yeah and then this is my next 45 90 over 45
135 180 and 45 225 plus 45 plus 45 plus 45 the equivalents in degrees sorry the
equivalence in radians an exact value would be I'm splitting this thing in a 1
2 3 4 5 6 7 8 into 8 all the way around is 2 pi so this is 1/8 of 2 pi I don't
know if I made sense this is going to be 1/8 of 2 pi and then to go in here and
this will give me PI over 4 so I'm counting by quarters 1/4 2/4 3/4 4
quarters five quarters six quarters all right
three over two is six fours so 5 4 6 4 7 fourths and then back to 8 4 8 forces 2
pi so those are the Radian the exact value 45 reference angles okay
well I hope that helped explain the difference between degrees and radians
hope that was useful and then I think the next video I'll go over more about
these exact values thanks for watching if you liked it hit subscribe I like to
hear your comments below thank you