www.mathcentre.ac.uk/.../5.4%20Radian%20Measurement.mp4

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.