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Degrees to radians

Sal converts the degree measures 150° and -45° to radians. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • male robot johnny style avatar for user Stijn le Page
    But why do we keep the pi? can't we do (150*pi)/180=2.6 radians ? or do you need to keep the pi?
    (41 votes)
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    • piceratops ultimate style avatar for user 1LEEJacob
      A radian is a relative unit based on the circumference of a circle. As you know, radians are written as a fraction with a π, such as 2π/3, 5π/4, or 3π/2. Since the equation for the circumference of a circle is C=2πr, we have to keep the π to show that it is a portion of the circle. Radian values can be used to calculate arc length using the radian and the radius multiplied together. Since it is encouraged to write these lengths in π units, The symbol is left give a π radian value.
      (28 votes)
  • mr pants teal style avatar for user Sabrina Keng
    At , Sal wrote that the circumference is 2pi radii, then at , Sal wrote 2pi radians:
    Is a radian equal to the radius. Or is it the same thing, just named differently? I mean, isn't the radian basically the radius bent around on the circumference. I'm really confused right now, so I would really appreciate it if someone would clearly explain this to me... the simpler the explanation, the better :) Thanks in advance!
    (16 votes)
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    • piceratops ultimate style avatar for user Pat Florence
      You're right about the way you visualize the definition of a radian. It is "the angle subtended by an arc equal in length to the radius." That is, one radian is the angle you would go through if you went one radius-worth of length on the circle. The radius, however, is a length measurement while the radian is an angle measurement.
      (31 votes)
  • purple pi teal style avatar for user DarthSho
    On my calculator , I have three angle notations, DRG. I know that D and R are degrees and radians respectively, so I checked on my calculator what it was. It turns out that it was gradians. So I looked it up and realized that a gradian is 10/9 a degree but I couldn't understand what it is used for. Could someone please answer my question? Thanks!
    (10 votes)
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    • leaf green style avatar for user kubleeka
      A gradian is 1/100th of a right angle. It was introduced when France was trying to make everything metric (they had a metric calendar too: it didn't work well). So they took the right angle, the "most natural" angle, and divided it into a hundred parts. So there's 400 gradians to a circle.

      So 400 grad=360° Divide by 360 and get 10/9 grad=1°
      (18 votes)
  • aqualine ultimate style avatar for user Satoshi Abe
    What does a question mean when is says to convert Degrees to Radians in terms of Pi?
    (5 votes)
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  • hopper jumping style avatar for user Aryan
    Is 2 pi radians equal to 2 pi radius?
    (6 votes)
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    • piceratops ultimate style avatar for user abassan
      They are different.
      Radians is a unit of measure like degrees. 2 pi radians, means you have 2 pi of something and they are radians. It is the same 360 degrees.
      A radius is the line from the side of the circle to the center. Sometimes radius refers to the lengh of that line. 2 pi radius means multiply 2 pi by the length of the radius which will give you half of the circumference. It is a length that changes depending on the size of the circle.
      (10 votes)
  • piceratops ultimate style avatar for user Neil
    Is there a symbol for radians?
    (6 votes)
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  • blobby green style avatar for user Robert Bolton
    question about radians. so if a radian is equal to about 57 degrees approximately (according to my book) how is the its measurements accurate to a full 360 degree circle?
    (4 votes)
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    • piceratops ultimate style avatar for user Just Keith
      There are 2π radians in a full angle (360°)
      1 radian is equal to 180/π which is about 57.2958°.
      It is easy to measure angles in radians. All you do is determine the fraction of a circle the angle sweeps out and then multiply that by 2π. For example, a right angle sweeps out ¼ of a circle. So ¼ * 2π = ½π
      (5 votes)
  • piceratops ultimate style avatar for user 1LEEJacob
    Overall, is the radian unit or degree unit used more? Does it vary over different subjects?
    (2 votes)
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  • aqualine ultimate style avatar for user Challenging Myself
    How can it be that 45°=45pi/180 radians while sin(45°)=sqr(2)/2? Can anyone help me to bring those two things together?
    (3 votes)
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  • male robot hal style avatar for user MRZasada
    Will a radian always equal the same. Like per say if i have another question asking how many radians are in 150 degrees will it always be the same or does it change per problem?
    (2 votes)
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Video transcript

- [Instructor] We're asked to convert 150 degrees and negative 45 degrees to radians. Let's think about the relationship between degrees and radians, and to do that, let me just draw a little circle here. So that's the center of the circle, and then do my best shot, best attempt to freehand draw a reasonable-looking circle. That's not, I've done worse than that. Alright, now, if we were to go in degrees, if we were to go one time around the circle like that, how many degrees is that? We know that that would be 360 degrees. If we did the same thing, how many radians is that, if we were to go all the way around the circle? We just have to remember, when we're measuring in terms of radians, we're really talking about the arc that subtends that angle. So if you go all the way around, you're really talking about the arc length of the entire circle, or essentially the circumference of the circle. And you're essentially saying, how many radius's this is, or radii, or how many radii is the circumference of the circle. You know a circumference of a circle is two pi times the radius, or you could say that the length of the circumference of the circle is two pi radii. If you wanna know the exact length, you just have to get the length of the radius and multiply it by two pi. That just comes from the, really, actually the definition of pi, but it comes from what we know as the formula for the circumference of a circle. If we were to go all the way around this, this is also two pi radians. That tells us that two pi radians, as an angle measure, is the exact same thing, and I'm gonna write it out, as 360 360 degrees. And then we can take all of this relationship and manipulate it in different ways. If we wanna simplify a little bit, we can divide both sides of this equation by two, in which case, you are left with, if you divide both sides by two, you are left with pi radians is equal to 180 degrees. How can we use this relationship now to figure out what 150 degrees is? Well, this relationship, we could write it in different ways. We could divide both sides by 180 degrees, and we could get pi radians over 180 degrees is equal to one, which is just another way of saying that there are pi radians for every 180 degrees, or you could say, pi over 180 radians per degree. The other option, you could divide both sides of this by pi radians. You could say, you would get on the left hand side you'd get one, and you would also get, on the right hand side, you would get 180 degrees for every pi radians. Or you could interpret this as 180 over pi degrees per radian. How would we figure out, how would we do what they asked us? Let's convert 150 degrees to radians. Let me write the word out. So, 150 degrees. Well, we wanna convert this to radians, so we really care about how many radians there are per degree, actually, let me do that in that color. (typing) We care about how many radians there are per degree. We'll do that same green color. Per degree. How many radians are there per degree? Well, we already know, there's pi radians for every 180 degrees, or there are pi... Let me do that yellow color. There are pi over 180 radians per degree. And so, if we multiply, and this all works out because you have degrees in the numerator, degrees in the denominator, these cancel out, and so you are left with 150 times pi divided by 180 radians. So what do we get? This becomes, let me just rewrite it. 150 times pi. All of that over 180, so this is equal to, and we get it in radians. And so, if we simplify it, let's see, we can divide the numerator and the denominator both by, looks like, 30. So if you divide the numerator by 30, you get five. You divide the denominator by 30, you get six. So you get five pi over six radians, or 5/6 pi radians, depending how you wanna do it. Now let's do the same thing for negative 45 degrees. What do you get for negative 45 degrees if you were to convert that to radians? Same exact process. You have negative, and I'll do this one a little quicker. Negative 45 degrees. I'll write down the word. Times, times pi radians, pi radians for every 180 degrees. The degrees cancel out, and you're left with negative 45 pi over 180 radians. So this is equal to negative 45 pi over 180, over 180 radians. How can we simplify this? Well it looks like they're both, at minimum, divisible by nine, nine times five is 45, this is nine times 20, so actually it's gonna be divisible by more than just, let's see... Actually, they're both divisible by 45. What am I doing? If you divide the numerator by 45, you get one. You divide the denominator by 45, 45 goes into 180 four times. You're left with negative pi over four radians. This is equal to negative pi over four radians. And we are done.