Introduction to radians Understanding the definition and motivation for radians and the relationship between radians and degrees
Introduction to radians
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- You are by now probably used to the idea of measuring angles in degrees.
- We use it in everyday language, we've done some examples on........
- where if you had an angle like that, you might call that a 30 degree
- 30 degree angle if you had an angle like this call that a 90 degree angle
- you'd often use a symbol just like that.
- If you were to go a 180 degrees, you'd essentially form a straight line.
- You go 360 degrees, you've essentially done one full rotation
- and if you watch figure skating on the olympics and someone does a rotation they say
- "Oh! they did a 360" or espescially in some type of some skateboarding competitions and things like that.
- But the one thing to realize is that it might not be obvious from the get-go
- but this whole notion of degrees, this is a human constructed system.
- this is not the only way that you can measure angles.
- And if you think about it yourselves,
- Why do we call a full rotation 360 degrees?
- And there's some possible theories and I encourage you to think about them,
- Why does 360 degrees show up in our culture as a full rotation?
- Well, there's a couple of theories there...
- One is ancient calenders and even our calenders close to this,
- but ancient calenders were based on 360 days in a year
- Some ancient astronomers observed that things seemed to move
- one 360th of the sky per day.
- Another theory is that the ancient babylonians
- liked equilateral triangles a lot and they had a base 60 number system.
- So they had 60 symbols, we only have 10.
- We have a base 10 and they had 60.
- So in our system we like to divide things under ten, they liked to divide under 60.
- So if you were to, if you had a circle and you divided it into 6 equilateral triangles.
- And each of those equilateral triangles you divided into 60 sections
- cause you have a base 60 number system.
- And you might end up with 360 degrees.
- What I want to think about in this video is an alternate way of measuring angles.
- And that alternate way, even though it might not seem intuitive to you from the get-go
- in some ways is much more mathematically pure,
- than degrees. It's not based on the cultural artifacts of base 60 number system
- or, astronomical patterns.
- To some degree an alien on another planet would not use degrees,
- especially if degrees are motivated by astronomical phenomena
- But they might use what we are going to define as a radian.
- Tere's a certain degree of purity here, Radians.
- So, lets just cut to the cheese and define what a radian is,
- so let me draw a circle here, my best attempt at drawing a circle.
- Not bad, and let me draw,
- centre of the circle and now let me draw this radius,
- and lets say that this radius,
- you might already notice that this word radius is very close to the word radian.
- And that's not a coincidence.
- So let's say that this radius, the circle has a radius of length r.
- Now lets construct an angle, I'll call that angle theta.
- So lets construct an angle theta.
- So let's call this this angle right over here theta and
- let's just say for the sake of argument that this angle is just the exact right measure,
- so that if you look at the arc that subtends this angle,
- That seems like a very fancy word.
- So let me draw the angle, so if you look at the arc that subtends the angle
- that seems like a fancy word
- and that's just talking about the arc along the circle that intersects the sides of theses two angles
- So this arc right over here subtends angle, this is the angle theta.
- So let me write this down, subtends this arc, subtends angle theta.
- Lets say theta's exactly the right size, so this arc is also the same length as the radius of the circle
- so this arc is also of length r.
- So given that if you were to find a new type of angle measurement and,
- you wanted to call it a radian which is very close to a radius.
- How many radians would you define this angle to be ?
- Well, the most obvious one, if you kind of view a radian as another way of saying radiusss or I guess a radii...
- Well, you say look, this is subtended by an arc of one radius.
- So why don't we this right over here one radian?
- Why don't we call this one radian?
- Which is exactly how a radian is defined,
- When you have a circle and you have an angle of one radian,
- the arc that subtends it is exactly one radius long.
- Which you can imagine might be a little useful as we start to interpret more and more types of circles,
- When you give in degrees, you have to do math and you have to think about the circumference and all of that
- To think about how many radiuses are subtending an angle.
- Here the angle in radians tells you exactly how many arc length is subtending the angle.
- So let's do a couple of broad experiments here.
- So, given that what would be the angle in radians if we were to go,
- so let me draw another circle here.....
- that's the center, start right over there.
- So what would happen if I had an angle, what angle if I were to measure in radian,
- what angle would this be in radian? You can almost think of it as radiussssss.....
- So what would that angle be? Going one full revolution,
- In degrees that would be 360 degrees. What would you, based on this definition,
- What would this be in radians?
- Well, lets think about the arc that subtends this angle is the entire circumference of this circle.
- Its the entire circumference of this circle.
- Well what's the circumference of a circle in terms of radiussss?
- So if this is length r, what's the circumference of the circle in terms of r?
- Well we know that it's gonna be 2 pi r.
- So going back to this angle, how many... whats the length of the arc that subtends this angle?
- How many radiusss? Well, it's two pi radiussss....
- Its two pi times r. So this angle right over here , this angle, I'll call this a different.. lets call this angle x.
- x in this case is going to be two pi radians.
- And it is subtended by an arc of length two pi radiussss....
- If the radius was one unit then this would be two pi times one, two pi radiusss...
- So, given that, lets start to think about how we can convert radians and degrees and vice versa.
- If I were to have, and we can just follow up over here.
- To do one full revolution, that is two pi radians, how many degrees is this going to be equal to?
- Well we already know this. A full revolution in degrees is 360 degrees.
- Well I could either write degrees or I could use this degree notation there,
- Actually let me write the word degrees. It might make thing a bit clearer that we're kind of using units in both cases.
- Now if we wanted to simplify this a little bit, we could divide both sides by 2,
- in which case we would get on the left hand side, we would get pi radians would be equal to how many degrees?
- Welll, it'd be equal to 180 degrees.
- 180 degrees and I could write it that way or I could write it that way.
- And you see over here, this is a 180 degrees and
- you also see if you were to draw a circle around here you've got halfway around the circle
- so the arc length or the arc that subtends the angle is half the circumference, half the circumference or pi radiusss...
- So we call this pi radians. pi radians is a 180 degrees.
- And from this we can come up with conversions.
- So one radian would be how many degrees?
- Well to do that, we just have to divide both sides by pi and on the left hand side you'd be left with one,
- I'll just write it singular now, one radian is equal to,
- I'm just dividing both sides, let me make it clear what I'm doing here just to show you it not voodoo.
- So, I'm just dividing both sides by pi here. On the left hand side we're left with one
- and on the right hand side we're left with 180 over pi degrees.
- So 1 radian = 180 over pi degrees.
- Which is starting to make an interesting way to convert them.
- Let's think about it the other way, if I were to have one degree, how many radians is that?
- Well, let's start off with, let me rewrite this thing over here.
- We said, pi radians is equal to 180 degrees.
- So now we want to think about one degree. So let's solve for one degree,
- we could divide both sides by 180.
- We are left with pi over 180 radians is equal to 1 degree.
- So pi over 180 radians is equal to one degree.
- This might seem confusing and daunting and it was for me the first time ......
- Specially because we're not exposed to this in our everyday life.
- What we're gonna see over the next few examples is that as long as we keep in mind,
- this whole idea that two pi radians is equal to 360 degrees or pi radians is equal to 180 degrees,
- which is the two things that I do keep in my mind.
- We can always re derive these two things.
- You might say, "Hey how do I remember if its pi over 180 or 180 over pi?"
- To convert the two things, well just remember and which is hopefully intuitive that two pi radians = 360 degrees.
- And we'll work through a bunch of examples in the next video,
- to just make sure that we're used to converting one way or the other. ENDSs.
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