Learn about Radians and degrees Video

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Learn about Radians and degrees

Learn about Radians and degrees Welcome to the presentation on Radians and Degrees. So you are probably already reasonably familiar with the concept of degrees I think in our angles modules we actually draw you to a bunch of problems. I mean you are probably familiar that a right angle is 90° or half of right angle or 45°. And you are also probably familiar with the concept that in a circle, that’s my best attempt in the circle. In a circle there are 360°. So today, I'm going to introduce you to another measure or unit for angles and this is called a radian. So what is a radian? So I'll start with the definition and I think this might give you a little intuition for what is even called the radian. So let’s say this is a radius of length are. A radian is the angle that subtends an arc and all subtend means is if this is the angle and this is the arc, this angle subtends this arc and this arc subtends this angle so one radian is the angle that subtends an arc. That’s the length of the radius. So the length of this is also r and this angle is one radian. Here you go. And I'm going to do this because I was wondering why they do radians. We all know degrees but actually we think by it actually and it makes a reasonable amount of sense. So let me use the line tool now. Let’s says that this radius is of length r and that this arc right here is also of length r then this angle is called θ is equal to one radian. Now, it makes sense that they call it a radian. It’s a kind of like a radius so let me ask you a question. How many radians are there in a circle? Well, if this is r, what is the whole circumference of the circle? It is 2πr, right? Well, you know that from the basic geometry module? So if the radian is the angle that subtends an arc of r, then the angle that subtends an arc of 2πr is2π radians. So this angle is 2π radians. If you are still confused, think of it this way. At an angle of 2π radian going all the way around subtends an arc of 2π radiuses or radii. I don’t know how to say the plural of radius. Maybe, it’s radians I don’t know. So why am I going through all this mess and confusing you? Well, I just want to one give you an intuition for why it’s called a radian and kind of how it relates to a circle and then given that there are 2π radians in a circle, we can now figure out the relationship between radians and degrees. So we said in a circle, there are 2π radians, right? Let’s say 2π radians. How many degrees are there in a circle? If we went around the whole circle, how many degrees? Well, that’s equal to 360° so there. We have an equation that sets up a conversion between radians and degrees so one radian is equal to 360 over 2π°. I just divided both sides by 2π which equals 180 over π°. Similarly, we could have done the other way. We could have divided both sides by 360 and we could have said 1° or I'm just going to divide both sides by 360 and I am clipping it. One degree is equal to 2π over 360 radians which equals π over 180 radians. So there, we have a conversion. One radian equals π over one - one radian equals a 180 over π° and one degree equals π over 180 radians and if you ever forget this, it doesn’t hurt to memorize this but if you ever forget it, I always go back to this that 2π radians is equal to 360° or another way it actually makes the algebra or the algebra I guess a little simpler is if you just think of a half circle. A half circle, this angle is a 180° right? That’s a degree sign or else right degree is out and that is also equal to π radians right? So π radians equal 180° and you can get the same now. One radian equals 180 over π° or one degree that’s degree is equal to π over 180 radians. So let’s do a couple of problems where you got the intuition for this. If I ask you 45° and to convert that into radians. Well, we know that 1° is π over 180 radians right? So 45° is equal to 45 times π over 180 radians and let’s see, 45 divided by 180 so this equals π over 4 radians, 45° is equal to π over 4 radians and just keep it in mind. These are just two different units or two different ways of measuring angles. And the reason why I do this is this is actually the mathematical standard for measuring angles although most of us are more familiar with degrees just from everyday life. Let’s do a couple of other examples. Just always remember this 1 rad = 180/π°, 1° = 180/rad. If you ever get confused, just write this out. This is what I do because I always forget whether it’s π/180 or 180/π so I just remember π radians is equal to 180°. Let’s do another one. So if I were to say π/2 radians equal how many degrees? Well, I already forgot what I had just written so I just remind myself that π rad = 180°. So π rad = 180°. I think you might get the point. Actually, let me just finish this problem and then I'll go attend to my wife. But we know that π rad = 180°, right? So 1 rad = 180 over that’s 1 rad = 180/π° right? I just figure out the formula again because I always forget it so let’s go back here. So π/2 rad = π/2 times 180 over π° and that equals 90°. I'll do one more example. So let’s say 30°. Once again, I forgot the formula so I just remember that π rad = 180°. So 1° = π/180 rad right? So 30° = 30 times π/180 rad which equals let’s see, 30 equal the 180 six times that equals π/6 radians. Hopefully, you have a sense of how to convert between degrees and radians now and even why it is called the radian because it is very closely related to a radius and if you’ll feel comfortable when someone ask you to deal with radians as opposed to degrees. I will see you in the next presentation.

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Learn about Radians and degrees

Welcome to the presentation on Radians and Degrees. So you are probably already reasonably familiar with the concept of degrees I think in our angles modules we actually draw you to a bunch of problems. I mean you are probably familiar that a right angle is 90° or half of right angle or 45°. And you are also probably familiar with the concept that in a circle, that’s my best attempt in the circle. In a circle there are 360°.

So today, I'm going to introduce you to another measure or unit for angles and this is called a radian. So what is a radian? So I'll start with the definition and I think this might give you a little intuition for what is even called the radian. So let’s say this is a radius of length are. A radian is the angle that subtends an arc and all subtend means is if this is the angle and this is the arc, this angle subtends this arc and this arc subtends this angle so one radian is the angle that subtends an arc. That’s the length of the radius. So the length of this is also r and this angle is one radian.

Here you go. And I'm going to do this because I was wondering why they do radians. We all know degrees but actually we think by it actually and it makes a reasonable amount of sense. So let me use the line tool now. Let’s says that this radius is of length r and that this arc right here is also of length r then this angle is called θ is equal to one radian.

Now, it makes sense that they call it a radian. It’s a kind of like a radius so let me ask you a question. How many radians are there in a circle? Well, if this is r, what is the whole circumference of the circle? It is 2πr, right? Well, you know that from the basic geometry module? So if the radian is the angle that subtends an arc of r, then the angle that subtends an arc of 2πr is2π radians. So this angle is 2π radians. If you are still confused, think of it this way. At an angle of 2π radian going all the way around subtends an arc of 2π radiuses or radii. I don’t know how to say the plural of radius. Maybe, it’s radians I don’t know.

So why am I going through all this mess and confusing you? Well, I just want to one give you an intuition for why it’s called a radian and kind of how it relates to a circle and then given that there are 2π radians in a circle, we can now figure out the relationship between radians and degrees.

So we said in a circle, there are 2π radians, right? Let’s say 2π radians. How many degrees are there in a circle? If we went around the whole circle, how many degrees? Well, that’s equal to 360° so there. We have an equation that sets up a conversion between radians and degrees so one radian is equal to 360 over 2π°. I just divided both sides by 2π which equals 180 over π°. Similarly, we could have done the other way. We could have divided both sides by 360 and we could have said 1° or I'm just going to divide both sides by 360 and I am clipping it. One degree is equal to 2π over 360 radians which equals π over 180 radians.

So there, we have a conversion. One radian equals π over one - one radian equals a 180 over π° and one degree equals π over 180 radians and if you ever forget this, it doesn’t hurt to memorize this but if you ever forget it, I always go back to this that 2π radians is equal to 360° or another way it actually makes the algebra or the algebra I guess a little simpler is if you just think of a half circle. A half circle, this angle is a 180° right? That’s a degree sign or else right degree is out and that is also equal to π radians right? So π radians equal 180° and you can get the same now. One radian equals 180 over π° or one degree that’s degree is equal to π over 180 radians.

So let’s do a couple of problems where you got the intuition for this. If I ask you 45° and to convert that into radians. Well, we know that 1° is π over 180 radians right? So 45° is equal to 45 times π over 180 radians and let’s see, 45 divided by 180 so this equals π over 4 radians, 45° is equal to π over 4 radians and just keep it in mind. These are just two different units or two different ways of measuring angles.

And the reason why I do this is this is actually the mathematical standard for measuring angles although most of us are more familiar with degrees just from everyday life. Let’s do a couple of other examples. Just always remember this 1 rad = 180/π°, 1° = 180/rad. If you ever get confused, just write this out. This is what I do because I always forget whether it’s π/180 or 180/π so I just remember π radians is equal to 180°.

Let’s do another one. So if I were to say π/2 radians equal how many degrees? Well, I already forgot what I had just written so I just remind myself that π rad = 180°. So π rad = 180°. I think you might get the point. Actually, let me just finish this problem and then I'll go attend to my wife. But we know that π rad = 180°, right? So 1 rad = 180 over that’s 1 rad = 180/π° right? I just figure out the formula again because I always forget it so let’s go back here. So π/2 rad = π/2 times 180 over π° and that equals 90°. I'll do one more example.

So let’s say 30°. Once again, I forgot the formula so I just remember that π rad = 180°. So 1° = π/180 rad right? So 30° = 30 times π/180 rad which equals let’s see, 30 equal the 180 six times that equals π/6 radians. Hopefully, you have a sense of how to convert between degrees and radians now and even why it is called the radian because it is very closely related to a radius and if you’ll feel comfortable when someone ask you to deal with radians as opposed to degrees. I will see you in the next presentation.

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