Stony Brook MAT 123 Fall 2015
Radian measure

Start   In this module we are going to learn about radians.
If you remember, degrees are there is 360 degrees in a circle.
Do you know why we have 360 degrees?
There is no special reason.
We use 360 degrees because lots of numbers go into 360 degrees.
1, 2, 3, 4, 5, 6, 8, 10, 9, 10, 12, ... If we had done 100 degrees it would be very hard to take a third of it.
So, 360 is a nice number and it has lots of divisors.
0:31The problem is we need another system of units which you will understand when you take calculus.
To be able to divide up the circle.
Actually that is not such a great picture.
Let us change that picture a little bit.
Let us just take a circle with a radius of 1.
So, how far is it around the circle.
Well, the circumference of a circle
1:00is 2 Pi times 1, so, it is 2 Pi.
So what we could do is we could say Some angle let's say that's 20 degrees.
How far is it around the circle?
And, that distance is the radian distance.
So, we would say, well, let us see.
20 degrees compared to 360 degrees, would be how far we are in radians, which we call x over 2 Pi.
1:30Do a little simplifying.
And, cross multiplying you get 2 Pi over 18, or Pi over 9 is x.
So, we would say is that the radian measure of 20 degrees is Pi over 9 radians.
2:00So, let us think about some simple conversions from degrees to radians.
Well, all the way around the circle is 2 Pi So, 360 degrees is is 2 Pi radians.
Half way around the circle, is 180 degrees is Pi radians.
So, you say wait a second, I thought Pi was 3.14 Well, that is the numerical value of Pi.
But, Pi radians is 180 degrees.
So if you want to convert
2:30from a number of degrees to a number of radians.
This is a very nice relationship to know.
Why, how would we use that?
Well, say if I want to convert from degrees to radians.
I multiply the angle
3:02by Pi radians over 180 degrees.
So, for example, 30 degrees is how many radians?
I take 30 degrees, I multiply it by Pi over 180, I do a little canceling, and I get Pi over 6.
So, 30 degrees
3:31is Pi over 6 radians.
If I want to go from radians to degrees, I multiply the angle by 180 degrees over Pi radians.
So, I just flip the ratio.
So, if I would have said, Pi over 3 radians
4:05is x degrees.
Then, I take Pi over 3 and I multiply it by 180 over Pi.
The Pi's cancel and I get 60 degrees.
So, 30 degrees is Pi over 6 radians.
4:31Pi over 3 radians is 60 degrees.
And, we, generally, don't write the words radians.
But, let's come up with some convenient angles, so, 30 (not 36).
45, 60 90, 180, 270, 360.
That's in degrees.
And, the angle in radians, well, 0 degrees is 0 radians.
30 degrees, you multiply by Pi over 180 radians, and we just showed that you get Pi over 6 radians.
What about 45 degrees?
Well, if you take 45, multiply by Pi over 180 and reduce
5:31You'll get Pi over 4.
60 we saw a minute ago was Pi over 3 90 degrees, 90 times Pi over 180 is Pi over 2 radians.
180 degrees is Pi radians.
270 times Pi over 180 will come out 3 Pi over 2 radians.
Or 3/2 of Pi.
And, 360 degrees we learned is 2 Pi radians.
6:01So, radians are equivalent to degrees in the sense of the part of the circle.
The radian is the distance along the circumference of the circle.
And, the angle is how big you open the opening between radii.
To get that length of the circle.
So, once again if you had a circle, the angle is the gap between the two radii.
The radians is how long the arc length
6:31is that corresponds to the angle.
So, that is how we learn to convert between radians and degrees.