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>> Julie Harland:
This is your math gal,
Julie Harland.
Please visit my website
at yourmathgal dot com
where all of my videos are
organized by topic.
This is radian measures
of angles, Part Three,
and we're going
to be converting
between degree measure
and radian measure.
So in my video radian measure,
Part Two, we determined
that 180 degrees equals pi,
or you could say pi radians.
And this is what we looked at.
We took the definition
of radian measure.
We know that if we did a full
revolution in degrees,
we call that 360 degrees.
But if this is a circle
with radius 1, we just say
that circumference is the same
thing as the radian measure,
and the circumference
of a unit circle is just 2pi.
So that gets us
to this important thing you
always want to remember
that 180 degrees is the same
thing as pi radians.
So, first of all,
does it seem reasonable
that 180 degrees is the same
thing as pi?
So let's look at this picture.
We know 180 is,
forms this straight line
from here to here,
and so this arc length,
does it make sense
that the length of that is pi
as a unit circle,
and remember,
if it's 2pi all the way
around, then from here
to over here, yes.
That would be half of that,
or 180 degrees.
So what about 90 degrees?
Now we're taking half of this,
right, half
of the 180 degree angle.
So we have this arc
length here.
So let's see.
Does it make sense
that you can get the answer
for whatever 90 degrees
by dividing each
of these by 2.
One hundred eighty degrees
by 2 gives you 90 degrees.
So pi over 2,
what do you write?
Well, you just write pi
over 2.
So 90 degrees is the same
thing as pi over 2 radians.
Now you don't have
to write radians.
You could just write the
number pi over 2.
Alright. What
about 45 degrees?
Well, that's half
of the 90 degrees
over here, right.
So you have 90 degrees.
If I divide it by 2,
that's like multiplying
by one-half, right, to get 45,
and so I have to take half
of pi over 2 also.
So that should be pi over 4.
Etc. Now what if I wanted
to know what 60 degrees was?
OK. Well, there's different
ways of doing it.
I could draw the picture
again, but if I know what 180
degrees is,
how could I get 60?
Well, so, in other words,
if I know that 180 degrees is
equal to pi, and I'm trying
to get 60, right,
I want to know what 60 degrees
equals, I could,
what could I do to 180
to get the 60?
I could divide by 3.
So 60 degrees must be pi
over 3.
But the thing is not everybody
could think of, oh,
how am I going to get
from 180 degrees to 60,
etc. So we're going
to learn how to convert
between degrees
and radians basically using
this important equation here.
A hundred eighty degrees is
the same thing as pi.
That means if I write pi
over 180 degrees,
pi is the same thing
as 180 degrees.
So that's just a number 1.
I could also write 180 degrees
over pi because, again,
180 degrees is the same thing
as pi.
So that is equal to 1.
So what I have is
that these are what I call
conversion factors here.
You could take anything
and multiply it by pi
over 180 degrees
or 180 degrees over pi
to convert from radians
to degrees
or to degrees to pi.
So the question is how do you
know which one to use?
Alright. So let's try
this problem.
Convert 45 degrees to radians.
So you've got 45 degrees,
right.
So here's the thing.
I want either multiply by pi
over 180 degrees,
or I want to multiply
by 180 degrees over pi,
and what do I want
to have happen?
I want not to have degrees.
So I need to multiply this
by the 1 so degrees
will cancel.
So this is my choice
right here.
So if I write this
as 45 degrees over 1, I,
if I multiply by pi
over 180 degrees,
then I see
that my degrees cancels.
If you're not used
to multiplying to, for,
in conversions, see my videos
on conversions.
Now what happens
when I do this?
The degrees cancel.
And so now I just need
to simplify by, you know,
canceling with a 45 and 180.
Now, 145 goes
into 18 four times, right.
So if I cancel this, that's,
the degrees cancel also,
right.
I'm going to get 1,
and that is 4.
So my answer here is going
to be, notice there's no
degrees anymore, pi over 4,
which is what we got
in the last page.
Right here,
notice that we also figured
out that 45 degrees is pi
over 4 by [inaudible]
of this picture here
and doing 180 in half, doing,
then doing 90 in one-half,
and we also got 45 degrees was
the same as pi over 4.
So that is how we convert
degrees to radians.
If I want to take radians,
and change it to degrees.
So now I want to take,
let's say pi
over 6 radians 2 degrees,
right.
So I start with pi over 6.
Now, I want to get my answer
in degrees.
So I want to multiply
by something so I get degrees
in my answer.
So the choice will have
to be 180 degrees over pi
as opposed to pi
over 180 degrees.
Think, if you're going
to degrees,
you want to put the degrees
in the numerator.
If you want to get rid
of the degrees,
you want to put the degrees
in the denominator.
So what happens here,
which is kind of nice,
is the pi's cancel,
and I've got 180
over 6 degrees.
So I'm going
to have 30 degrees.
So pi over 6 is the same thing
as 30 degrees just
like [inaudible],
and pi over 4 was equal
to 45 degrees,
etc. And we'll do some more
problems converting back
and forth on the next video.
This is your math gal,
Julie Harland.
Please visit my website
at yourmathgal dot com
where all of my videos are
organized by topic.