In this video, we're going to look at a different way
of measuring angles.
Well aware. In a circle.
There are 360 degrees all the way around.
360 degrees in a circle, 360 degrees all the way around. What
we need is a new way of measuring angles and this is
needed because when we look at trig functions, particularly
their differentiation, we will need a different measure of
angle from degrees. So let's have a look at what that measure
is. So we take
a circle. Radius R.
And I want to mark off on the circumference a length that is
actually equal tool are so that distance around the
circumference is equal to R, the radius. Now we join up.
There. That's our and this angle here theater we describe
as being one Radian.
So Theta equals 1.
Radian
And if we want a definition of that, then the definition is.
The angle of one Radian is the angle subtended at the center of
the circle by an arc that is equal in length to the radius of
the circle. So that gives us our definition of a radium.
Let me just draw.
3. Circles.
Let's put that definition into effect again, where we've got.
A distance equal to R, the radius of the circle around the
circumference. And there's our theater equals 1.
Radian now let me.
Go round the circumference a distance of two are.
As the radius again, let's take half way around there so that
this is Theta. In here is one Radian, an theater in here
is also one Radian.
So what we can see there is that this distance to our
is in fact two radians times by the radius, and so that
gives us a general formula.
That if we have any angle theater subtended at the center
of the circle then the arc length S around there is given
by S equals R Theater.
And that's the arc
length. And it's true if theater
is in radians.
So that gives us a very good way of calculating the arc length
when we know the angle subtended at the center, and we know its
value in radians.
So. Can we get an equivalence
between our radiant measure and
our degrees? Well, let's have a look at a circle again.
The circumference of this circle.
We know by the formula to
Π R. Now let's compare that with our
formula arc length.
Our formula for arclength we know, is our theater.
Well, he is, AH.
And here is our.
And what that means is that for this circle, the distance all
the way round is 2π, R, and therefore the angle.
In radians all the way round is this number here 2π.
So what we've got is that 2π radians.
Is exactly the same as 360 degrees.
Now this equivalence enables us to write down a whole series of
Equalities. Between angles, so we've got 2π radians.
Is the same as 360 degrees, so Pi radians is
the same as 180 degrees. Just dividing both sides by
two. If we divide by two, again π by two radians.
Notes, so I said that Π by 2 not π / 2, but pie by two. It's a
shorthand, so Pi by two radians is 90 degrees, and that's an
important one, because that's a
right angle. Pie by 4 / 2 again high by
4 radians is 45.
Degrees. So these are the important angle, so to speak.
Some other quite important angles in trigonometry are 60
degrees, well, 60 degrees is 1/3 of 180 degrees. So Pi by three
radians. Is the same as 60 degrees, another angle that's
important is the angle 30
degrees. And 30 degrees is 1/6 of 180 degrees, so π
by 6 radians.
Is the same as 30 degrees.
OK, we've got all of these equivalences, but the one thing
we haven't got is how big is a Radian in terms of degrees. One
Radian equals how many degrees?
Or will start with this relationship here that Pi
radians is 180 degrees.
So. Π radians
is 180
degrees. So
one Radian. Must be 180 degrees divided by
pie and if we use a Calculator to do that,
we find that the answer is 57.296 degrees.
2 three decimal places now it's actually rare that we need to do
this particular calculation, but what it does do is it gives us
some idea of exactly how big a
radiant is. Now.
So far I've written down the word Radian in fall.
But I haven't written down the word degrees in full. Have used
this symbol. This little zero up here to signify degrees. So what
do we do about radians? How can we signify Aready? And so if we
write down 1.5 radians, how we able to tell that this is
radians? One way is to put a little see up there at the top,
usually in italic.
Print if it's published in a book, so 1.5 radians. Another
way is to write 1.5 ramps where Rams is obviously short for
radians. What we've done does not affect trig ratios. So for
instance, the cosine of π by three is exactly the same as the
cosine of 60 degrees because the two angles are the same, the
tangent of Π by 4 is exactly the same as the tangent of 45
degrees. And the sign of three Pi by two is exactly the same
as the sign of 270 degrees.
One thing to notice here is that because I've got pies in.
I haven't written down a symbol.
Because one, we've got a pie in the angle it's taken as read
that these are measured in
radians, so. We've got our formula S equals R Theater,
where S is the arc length and theater is in
radians. What's the formula for S if Vita is in degrees? Let's
just have a look at our circle.
Yeah.
Radius R. Our arc length S is around the
circle like so.
Radius R and this angle in here let's market as
being theater degrees.
So what's our formula? Well, S must be a certain
proportion of the whole
circumference. S over 2π, R and the angle theater will be
exactly the same proportion of 360, and so we can calculate
the arc length S as 2π R that's multiplying both sides
by 2π. R times Theta over
360. So there we've got our arc length formula for Theta in
radians. And here's our arc length formula for theater in
degrees. Notice how much simplier this one is? What about
another formula? What about the area of this sector?
Well, let's have a look at
that. Is our circle again?
Radius R. Angle
theater It's called the center of
the circle. Oh, and let's label the two points A&B.
What's the area of this sector of the circle?
This slice of the pie, so to speak. What's its
area? Well, the area?
Of the sector.
As a proportion, or a fraction of the area.
Of the circle.
Must be the same fraction, the same proportion as the
angle is of 2π.
All the way around. So the area of the sector considered
as a fraction of the whole circle must be in the same
ratio as the angle.
Considered as a fraction of the whole angle all the way round in
radians. Well, we can work this out area.
Of the sector.
And we have a formula for the area of the circle. We know
that's π R squared.
Equals feta over 2π, And if we multiply it by this expression
here, the Pi R-squared the area of the circle. Then we have the
area of the sector.
Is equal to π R squared times theater
over 2π and the
pi's cancel. That leaves me with
a half. All
squared Theta. So for theater in radians this is our
formula. The area of a sector is 1/2 R-squared Theta.
OK, we've got some formula here. We've got some relationships.
Let's have a look at how we can use them to do some
calculations. So in our first
example. Let's take a circle.
Of radius. 10 units
And let's take an arc length.
Of 25 units.
Now if I make this a little more representative, 25 more likely
to be around there 10 and 10, and the question is what's this
angle here in terms of radians and also in terms of degrees?
So. What's our formula for Arclength? That's arc length
equals are times by theater. If theater is in radians.
So as we know to be 25.
We know to be 10 times by theater, so theater is 25
over 10 which is 2.5 and I can put a little see
up there to stand for radians. Or I could write it
as 2.5 rats.
So that's our angle in terms of radians. What will our angle
be in terms of degrees? Well, one way might be to convert
this answer here. These two point 5 radians into degrees.
Let's remember that we had Pi
radians. Was
180 degrees. So one Radian
was 180 / π degrees, so
2.5 radians.
Must be 180 over π times
by 2.5. And we can use a Calculator to work
this out. And the answer is
143.2 degrees. Now let's have a look at another example.
Start from the same point. That is, we've got a circle.
We've got circle of radius 10 centimeters, and this time let's
say we're going to take an arc length of 15 centimeters.
The question is, what is the angle in radians? Here at the
center, but also, what's the area of this sector oab? So
let's begin with our formula for arc length S equals R Theater.
Our arc length is 15 hour, radius is 10 and the angle
is theater. So we divide both sides by 10 to give us
theater is 15 over 10, which is 1.5 radians.
Now area of the sector. We have a formula for that.
Area of sector. We know that that is
1/2 R-squared Theta.
So we can take our values. We know that R is 10,
so that's 10 squared for the R-squared, times by angle we've
just calculated 1.5.
So that's 1/2 times by 100 times by 1.5 now 100 times
by 1.5 is 150 and we want half of it, so that
is 75 centimeters squared. So that's the area of this sector.
Further questions we could ask is what's the area of the
segment the minor segment? If you take a line across a circle,
it divides a circle into two parts, a big bit, the major
segment and a smaller piece. They minus segment, and we can
ask what's the area of that
minor segment? Let's just draw another diagram and look at that
again, 'cause it was quite a lot of vocabulary in that quite a
lot of words. So we're going to take a line across the circle.
We're keeping this as 50. We've got the center of the circle and
we know what the angle is, so there's our as are we know that
to be 10.
And we've calculated what this angle is. So here's the line
that we drew across the circle, and it divides circling two
parts of big part like that which we call the major segment
and a small part like that which we call the minor segment.
The question is, what's the area of this minor segment? How can
we work it out? Well, we know the area of the sector
of the circle area.
All sector 'cause we calculated that and that was 75
centimeters squared. What we need to be able to do is
workout the area of this
triangle. Area of the triangle AOB. The area of the
triangle will be 1/2.
All squared sine theater.
So that's 1/2 times by 10 squared times by the sign of 1.5
radians. And again we can work this out on a Calculator. We
have to be very careful that are Calculator a set up to work in
radians, not in degrees, to work in radians. And so here we have
1/2 times by 100.
Times by and the sign of this angle is nought. .997 and then
lots of decimal places. And if we work that out on a Calculator
you get 49.8747 and so our final answer, which we want is the
shaded area here area.
All the minor.
Segment. And that's got to be the difference between
the area of the sector.
And the area of the triangle so
that 75. Take away 49.8747 and to a
suitable degree of accuracy 2 decimal places. That's
25.13 centimeters squared.
Let's have a look at one final example. We have converted
radians into degrees, but one of the things that we haven't done
is to convert an angle that's in degrees into radians.
So let's take an angle 120 degrees. What is this
angle in radians? Let's start off with our relationship that
Pi radians is 180 degrees.
Now we were able to say that one Radian was 180 over
π, dividing both sides by pie. So if we want 1 degree,
then we can divide both sides by 180. So π over 180
radians is equal to 1 degree. We want 120 degrees, so π
over 180. Times 120 is equal to 120 degrees and again we
need a Calculator to work this out. And when we work it
out we get 2.09 radians and that's to two decimal places. Is
the same as 120 degrees.
Important things to take away from this video are this
relationship. Π radians is 180 degrees and the two formally
that we had.
One for the arc length, which was S equals R
theater and one for the area of the sector which
was area of the sector is equal to 1/2 R-squared
Theater and both of these to apply when theater is
in radiance.