BAM! Mr. Tarrou! In this math lesson we're going to be looking at 4 examples dealing
with angle measures that are in the unit measure of degrees, minutes, and seconds. Minutes
and seconds are a part of a subdivision of a degree. Like a decimal. And most of the
work with angles in our trig book, unless you're you know working with unit circle problems
and you have to memorize those 16 angles around the unit circle, are going to have these angles
measured in what's called a decimal degree. But there will be a few occasionally through
the book that are in a different unit of measure. You know, with this DMS notation. Instead
of having a part of an angle as a decimal, we'll break it down by minutes and seconds.
So we have to know how to deal with that. So many angle measurements are not single,
whole number values. Now that I'm starting to read this - what is in this lesson - I'm
gonna have links, time stamps, in this video's description letting you skip ahead. We're
gonna do one example where we convert from decimal degrees to DMS. Another example, we
work the other way around, DMS to decimal degrees. And our last two examples are going
to be addition and subtraction of angles when they're in this degree, minute, and second
notation. Okay. So, like centimeters are a part of a meter and millimeters are part of
a centimeter. A minute is part of a degree and a second is a part of a minute. So just
taking these whole measurements of degrees and breaking them down more and more and more
to increase sort of, or decrease that measure of accuracy to get, well, sort of like a more
accurate measurement of a particular angle. Now DMS notation. We got one degree is equal
to 60 minutes. So If we're looking at minutes as being a part of a degree, there's 60 parts
in a degree if those parts are a number of minutes. And so thus one sixtieth of a degree
is equal to one minute. And this is the single little hatch mark or hash mark, whatever that's
called, that is sort of a symbol for the word minute. That's why I had this underlined in
yellow to... I'm not sure how much different that's showing up on the camera , maybe I
should have picked a different color but that's orange, that's yellow. And minutes. Symbol
notation. A single little hash mark on the upper right hand corner. Now, if breaking
a degree own into 60 parts isn't, you know, giving you an accurate enough measurement,
you can then take those minutes and split them up again. And one minute is equal to
60 seconds. So, one sixtieth of a minute is one second. And so thus one second is equal
to 1 three thousand six-hundredth or one thirty-six hundredth of a degree. Very, very minute parts
of an angle measure. And maybe you need to break these degrees down into that small of
an increment maybe for astrology or well literally rocket science and some measurements like
that that require a super precise unit of measure. Okay. So that means that if you want
to convert from DMS notation - degree, minute, and seconds - to a decimal degree, more like
we'd be used to with the calculator. We're going to take the number of degrees, we're
going to - that's going to be like the whole number value in the front - take the number
of minutes and divide by 60 and take the number of minutes and divide it by 3600 and add up
those three values and that is what we're going to use in our second example. But first,
we're going to convert this decimal degree to the unit of measure of degree, minutes
and seconds. We have 27.614 degrees. Okay. Now let's see here. This is of course a whole
number value. We have 27 degrees. And then we have this part of a degree, right? Well
that's what minutes are. It's a part of a degree. And there's 60 minutes in a degree
so you know really I guess if I wanted to know how many minutes total, I could just
do 27.614 and multiply by - the whole thing - by 60 and have this HUUGE number of minutes.
But you know I don't wanna break down.. I don't wanna have everything in terms of minutes.
I just, I have a nice whole number value of 27 degrees. We're going to take the point.
We're going to take, add a plus sign, and take the .614, make sure you take your decimal
with it, this part of a degree and we're going to breakdown into the next smaller unit of
measure which again are those minutes. We're gonna do that by doing .614 times 60 and when
we do that, right off the top of my head, we have... 27 degrees plus .614 times 60 is
36. 84 minutes and again that's because one degree is equal to 60 minutes. That's what's
allowing me to do that conversion. Just like if you were taking the number of feet, a larger
measurement, and you wanted it to be a number of minutes or - excuse me - inches, you would
multiply by 12. Taking a large measurement and finding out how many smaller parts are
in there like 12 inches in a foot. You would multiply by 12 to find inches in a feet, and
so thus we're taking .164, this part of a degree, and multiplying it by 60 so we can
get the number of smaller of units that are in that sort of larger unit of measure. So
now well we have another decimal and we did of course say that we can break minutes down
even more by going into seconds. So we have a whole number of minutes so now we're at
27 degrees and 36 minutes, but then we have an ability to go one more step and turn this
..84 minutes into a number of seconds. Well, there, y'know, are 60 secONDS in a minute.
So we're gonna do this breakdown again and do .84 times 60 again and that's because like
I have written here, one minute is equal to 60 seconds. And that multiplication again,
right off the top of my head, comes out to be that, well, let's write the whole thing
out, 27 degrees, 36 minutes, and then finally .86 - well that kinda is a bit sloppy - .84
times 60 comes out to be 50.4 seconds. Now we still have a bit of a decimal in there,
but we're not gonna break this angel measure down into any smaller units of measurements,
anything smaller than seconds. So that is the end of my first example and 27.614 degrees,
out of your calculator does come up to 27 degrees, 36 minutes and 50.4 seconds. By the
way, if you have a TI 83 or 84 in blue or yellow depending on what calculator you have,
above one of those buttons it says angle. So if you do second - angle, you'll have a
menu pop up and one of those menus will have a little arrow in it that looks something
like this. It'll have DMS of degree, minute, seconds and it'll do that conversion for you
so can check your answer. I am sure you teacher is going to want to see that work written
out by hand, not just putting that into your calculator. But it is a nice way to check
it if you are allowed to use your calculator during your quiz or of course during your
homework to make sure that you are doing these correctly. Don't practice these problems not
checking your answers and teach yourself how to do it the wrong way if you are answers
are incorrect. Make sure you check each one as you do the problems so you practice them
the right way. Now we did this conversion from decimal degrees to DMS, let's turn them
around. Nananananana.... Convert this angle which is in DMS format, degrees... minutes...
and seconds, to decimals. Well that example that I just erased, we had the decimal version.
We multiplied by 60 to find the number of minutes and a part of those minutes we multiplied
by 60 again to get the number of seconds. Now I don't want to take my seconds and convert
them into minutes and then convert them into the degree measure. So I am going to do what
I have up here. I am going to take the degrees and leave it out front just like I did with
the other conversion. I am going to remember that the other way around we multiplied...right?
We are going to... We were converting that decimal into DMS format. Well now I have this
part, this 13 minutes, which is a very small measurement. It is 1/60th of a degree. Because
ultimately I just want a degree measurement. So if I want to go from a really really small
measurement and get to a larger measurement, I need to do what? Let's see here. Let's take
again that inches and feet thing. If I have 24 inches, those are relatively small units
of length, and I want to convert the number of inches to a number of feet...which is a
longer unit of measure... 24 inches is... Well 24 divided by 12 is 2 feet. So when you
are going from a really small measurement to a larger unit of measure you want to divide.
Thus... I am just trying to explain here why I am taking the number of minutes and dividing
by 60, besides the fact it is just the opposite of what I did in the previous example. So
we are going to take 13 minutes and we are going to divide that by 60 because there is
60 minutes in a degree. So we are taking the number of minutes and dividing by 60, like
I... I am repeating what I just said a minute ago. Now we are going to take the 42 seconds
and I want to go straight into degrees now. Because I want to go right into the decimal
format. I want to divide by 3600 because again, the other way around, we multiplied by 60
to get the part into minutes and then we multiplied by the part of a minute into a second. So
multiplying by 60 times 60 is 3600. So going in the reverse order we are dividing. Alright.
So that comes out to be, right off the top of my head, 125 plus, 13 divided 60 is equal
to .216667... not this starts to repeat and our final answer is going to be just a little
bit off from where we started from because of this round off error. We get this .6 repeating,
so I am going to just go to a few decimal places there and then round up to seven. Then
42 divided by 3600, that comes out to be .011667. Putting that all together we have approximately,
and it is approximate. This is exactly 13 and this is exactly 42, but trying to convert
into that decimal format we have these run on decimals and so our final answer is going
to be close but not exactly right. So the answer is approximately 125. 2283 degrees.
Now if you put this into your graphing calculator and do second-angle and hit that convert to
DMS format, this is not going to get exactly back to this 125 13 minutes and 42 seconds
again because of that round off error. You can run these 6's a little bit farther out
and you will be closer. That is how you convert from degrees minutes and seconds into a decimal
measurement of an angle. Now how do you add angles and subtract angles that are in this
DMS format? Let's find out. Nanananana... Example three. We are going to add 63 degrees
and 41 minutes to 28 degrees and 34 minutes. Ok, well you could also write this as 63 degrees
and 41 minutes, if you prefer, and 28 degrees and 34 minutes. Only I can't of this as sort
of the one's, ten's, and hundred's columns because this is not really base ten, it is
base 60 because there is 60 minutes in a degree... and another 60 seconds in a minute or 3600
seconds in a degree. So we will have to do a little carry over type work but it is going
to be different because we are not in base ten. Let's not worry about doing any carry
over and actually just add these columns up, the degrees and the minutes and see what we
get. 41 plus 30 is 71, and 71 plus 5 is... Thinking of the answer. Let's try that again.
41 plus 30 is 71, and 71 plus 4 is 75 minutes. And we have 63 plus 20 is 83, and 83 plus
8 is going to be 91. Ok. So we have 91 degrees and 75 minutes. Well, you know, if asked someone
how tall they were and they said, oh..you know.. I am five feet and 13 inches. That
does not make any sense. Why would you say 13 inches? Why don't you just say 6 feet and
1 inch. Why would you have a part that is bigger than the whole. That is what we need
to worry about. So here...let's see. How can we do this? This is a few ways you could show
this. I could say, I am going to subtract out 60 minutes which is going to give my 75
minutes... take away 60 minutes is going to be 15 minutes. But, you know... What I am
going to do? I can't just do that. I am changing the value of my answer. So if I take out a
60 minutes, I need to add back an equivalent of 60 minutes which is 1 degree. Thus our
final answer is...I don't know why I put it all the way over there... 92 degrees and 15
minutes. There is other ways you could show that as well. But by subtracting 60 minutes
and adding in another degree I am taking away and adding an equivalent value so I am basically
adding by zero if you will. I am sure there is maybe some other notations that maybe your
teachers will show to that taking out of the 60' and bring in the extra 1 degree. But I
hope that works for you. Find the complimentary angle to 30 degrees 15 minutes and 40 seconds.
Well we have to understand what the definition means for complimentary angles. Hopefully
you do from Geometry. Complimentary angles are two angles that add to 90 degrees. Supplementary
angles, not that this is in this particular question, but supplementary angles are two
angles that add up to 180 degrees. Just some vocabulary you need to know. You might know
how to add and subtract angles that are in this DMS format, but if the question gives
you some vocabulary and you don't know what it means...you are not going to be able to
get the question right. Do you know that means that they add up to 90 degrees? So we are
going to do 90. I hope so otherwise you can't do this question. So we are going to do 90
take away 30 degrees 15 minutes and 40 seconds. Well, just like if we were doing regular subtraction,
except we are not in base ten. If I was doing 200 minus... oh I don't know...nine. I would
need to borrow from the 2 and make this 10, then I would have to borrow from the ten and
add that over and make that a 9 and then bring a 10 over...or a 1 over for the hundreds,
10's units, and one's place. I am borrowing and bringing over 10 because this is a base
ten number system and that would be 191. 10 minus 9 is 1, and you just bring the 9 and
the 1 down. Well we are not borrowing on a base ten, we are borrowing on a base 60. So
I am going just take this in really really small steps and say....well I don't have anything
to take the 15 minutes away from. So let's borrow a degree and make that 89. Taking away
a degree let's me add in 60 minutes. I am getting a little bit sloppy so lets write
that as 89 degrees and 60 minutes. Awesome! Now I don't have anything to subtract my 40
seconds from. So we need to borrow a minute, which is going to bring that down to 59 minutes
and add in it's equivalent which is 60 seconds. But again I am trying to keep this as neat
as possible. I probably should be using some different colors in here. How about we do
this. It is bugging me a little bit. Right now you are hitting the fast forward button
on the lesson. Alright. Now bringing this over here we have 89 degrees. We have 59 minutes.
Then we have 60 seconds. We are taking away finally now we can do the subtraction of 30
degrees 15 minutes and 40 seconds. Our final answer is going to be 60 minus 40 which is
20 seconds. 59 minus 15. Well 59 minus 10 is 49, and 49 minus 5 is 44 minutes. Finally
89 minus 30 is going to be 59 degrees. Let me just make sure that I am agreeing with
my answer that I have already worked out. Yes I am. That is how you deal with degree
minute second notation, or DMS notation for measures of angles. I am Mr. Tarrou. BAM!!!
Go do your homework:D
With a remainder of 2. 2/5, I can't believe I am pausing trying
to write that answer out. So 7 divided by....Oh for good Lord!