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1. I'm having a really hard time understanding why you multiply by pi/180 to get degrees. I completely don't understand why this is used, and how you arrive at this. I know that pi radians = 180 degrees but I don't understand how this is used to convert into radians/degrees. I originally thought this was used by rearranging but even then it doesn't make sense to me. This is really stressing me out as its a really fundamental point within this topic and I don't understand it! All help is really appreciated so thank you in asvance
2. (Original post by JakeRStudent)
I'm having a really hard time understanding why you multiply by pi/180 to get degrees. I completely don't understand why this is used, and how you arrive at this. I know that pi radians = 180 degrees but I don't understand how this is used to convert into radians/degrees. I originally thought this was used by rearranging but even then it doesn't make sense to me. This is really stressing me out as its a really fundamental point within this topic and I don't understand it! All help is really appreciated so thank you in asvance
pi/pi radians -> 180/pi degrees (divide both sides by pi)
x radians -> 180/pi * x degrees
3. (Original post by JakeRStudent)
I'm having a really hard time understanding why you multiply by pi/180 to get degrees. I completely don't understand why this is used, and how you arrive at this. I know that pi radians = 180 degrees but I don't understand how this is used to convert into radians/degrees. I originally thought this was used by rearranging but even then it doesn't make sense to me. This is really stressing me out as its a really fundamental point within this topic and I don't understand it! All help is really appreciated so thank you in asvance

Degrees are an arbitrary way of dividing a circle. I hope it easy to see that an angle in degrees is converted to a fraction of a circle by dividing by 360.

With radians, we have decided to define the circle as units rather than 360 and so a particular angle in radians is made into a fraction of a circle by dividing by .

So first of all turn whatever you have to a fraction of a circle by dividing by 360 if you have degrees or by if you have radians. Once you have done this, you just multiply up by the other one. I.e by if you started with degrees or 360 if you started with radians.

I think that the confusion that candidates have with radians often arises because they spend too much time trying to use a formula rather than working from the fundamentals. The relationship between 180 and the angle used less obvious than with 360. The 180 only arises because of the cancellation of the 2 in with 360.
4. (Original post by JakeRStudent)
I'm having a really hard time understanding why you multiply by pi/180 to get degrees. I completely don't understand why this is used, and how you arrive at this. I know that pi radians = 180 degrees but I don't understand how this is used to convert into radians/degrees. I originally thought this was used by rearranging but even then it doesn't make sense to me. This is really stressing me out as its a really fundamental point within this topic and I don't understand it! All help is really appreciated so thank you in asvance
360 degrees in a circle (by definition)
2 pi radians in a circle (by definition)

So 360 degrees = 2 pi radians
(divide by 2)
5. (Original post by nerak99)
With radians, we have decided to define the circle as units rather than 360 and so a particular angle in radians is made into a fraction of a circle by dividing by .
I think that this is a rather back-to-front way to express it. For a circle of radius , a radian is usually defined as the amount of angle subtended by an arc of length at the centre.

So imagine measuring a piece of string of length and wrapping it around the circumference of the circle, then draw radius lines from the centre to the ends of the arc - the amount of angle between those two radii is 1 radian. Then, we can say that since the circumference is , then we can fit exactly arcs of that size around the circumference, and hence since each of those arcs subtends 1 radian, we get radians of angle around a circle.
6. (Original post by atsruser)
I think that this is a rather back-to-front way to express it. For a circle of radius , a radian is usually defined as the amount of angle subtended by an arc of length at the centre.

So imagine measuring a piece of string of length and wrapping it around the circumference of the circle, then draw radius lines from the centre to the ends of the arc - the amount of angle between those two radii is 1 radian. Then, we can say that since the circumference is , then we can fit exactly arcs of that size around the circumference, and hence since each of those arcs subtends 1 radian, we get radians of angle around a circle.
You are of course quite correct. I am sure that the OP will find your post illuminating My primary objective was to give them some tools to cope with the difficulties they were describing. What you say is accurate and reveals a great deal about the origin of radians rather than addressing the student's problem.
7. (Original post by nerak99)
You are of course quite correct. I am sure that the OP will find your post illuminating My primary objective was to give them some tools to cope with the difficulties they were describing. What you say is accurate and reveals a great deal about the origin of radians rather than addressing the student's problem.
Well, in a sense, you are correct too, since we *could* define a radian in the way that you did, and everything would be fine, but it isn't standard.

I suspect that the OP's problems stem from the fact that they aren't comfortable with direct proportion.
8. 2pi/360
9. Well, Our OP now, has both to choose from. Assuming he was introduced to radians sung the standard method, clearly the lesson was not a roaring success.
10. (Original post by nerak99)
The 180 only arises because of the cancellation of the 2 in with 360.
Which is why tau is much better.
11. (Original post by constellarknight)
Which is why tau is much better.
Help me plz
12. (Original post by Naruke)
Help me plz
13. The thing about have a multiple number of pi turning up is that pi then cancels out in a large number of calculations.

Also, if you plot a sin between 0 and 2 pi, the gradient of the curve at each point exactly match a Cosine. That way we can happily say that the differential of sin is cos. If we did not work in radians for calculus a whole bunch of factors of 2pi would crop up all over the place.

In addition, all our series expansion would have pi all over the place. There really is no question about it that having angles in radians is a good thing.

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