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I’ve been messed over on a test (at least one that I can clearly remember, probably at least a few more) (and it was only a few points, but still very frustrating) because I forgot to switch my calculator from degrees to radians when going from physics class to calculus.

Why do they use different measurement systems? Wouldn’t it be best to mainly use one system? What is better about either one that makes them preferred by that class? Specifically, why are radians used in some physics problems while others use degrees?

Qmechanic
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Doragon
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    https://math.stackexchange.com/q/1797756/ – Brain Stroke Patient Mar 18 '21 at 20:28
  • Hello! I have made the attempt to edit your question to make it more suitable for Physics SE (instead of pure Mathematics). If you don't like it, feel free to rollback. Thanks! – jng224 Mar 18 '21 at 20:57
  • Thanks! I didn’t know that some physics problems used radians. – Doragon Mar 18 '21 at 20:58
  • Possible duplicates: https://physics.stackexchange.com/q/33542/2451 , https://physics.stackexchange.com/q/252288/2451 and links therein. – Qmechanic Mar 18 '21 at 21:46
  • You have just hit upon the exact reason that you should work extra problems, outside the class room, before a test. Those extra problems uncover lack of knowledge, poor habits, oversights, etc., and it is urgent that you make note of such things and internalize them to the point that you quit making the same mistakes over and over. – David White Mar 18 '21 at 22:36
  • @Doragon, one more comment. EVERY time that I intend to use a trig function on my calculator, I first enter SIN(30). If the calculator is in degree mode, I get an answer of 0.5. If I don't get this answer, I know that the calculator is in radian mode. This little habit keeps me from working problems in the wrong degree mode, and I think you would find it helpful too. – David White Mar 18 '21 at 23:39
  • @DavidWhite great advice!! The main reason I keep forgetting is that I’m too lazy to dig out of my calculator what setting it’s in, and since I don’t always change it in the first place (we don’t always need degree equations) I usually forget what mode it’s in. – Doragon Mar 19 '21 at 00:04

3 Answers3

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Degrees measure angles, while radians actually measure distance (in some sense). This is because a radian is defined as

the angle subtended from the center of a circle which intercepts an arc equal in length to the radius of the circle

(Source)

as can be seen in this animation:

enter image description here
(Source)

This makes the radian particularly useful when talking about rotational motion. For example, saying that you moved a distance of 2 radians makes a lot more sense than saying you moved a distance of, let's say, 30°.

So if you encounter radians, there is most often a (hidden) circle. This is also why radians are most commonly used for descibing trigonometric functions. On the other hand, describing the angle between a light ray hitting a mirror and the surface would usually be given in degrees.

This answer may be a bit unsatisfactory since you could always convert radians into degrees and back and you would be right, but this is the best that I can explain it. I'll leave you some links for further reading:

Radians vs. Degrees - physicsthisweek
Degrees vs. Radians - mathwithbaddrawings

jng224
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    Re, "...in some sense." The "sense" is that radians measure the ratio of the arc length to the radius of the circle. – Solomon Slow Mar 19 '21 at 00:24
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Radians is the "big boy" units to use. Degrees are the "baby method" to use. Mainly because they are defined such that there are $2\pi$ radians in an a circle. Meaning $\theta/360 \cdot \pi \cdot r^2$ becomes simply $1/2r^2 \theta$.

$\theta/360 \cdot2\pi\cdot r$ simply.becomes $r\cdot\theta$

It is the most natural units to use.

Degrees were invented due to their easy divisibility.

jng224
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jensen paull
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You could certainly use degrees if you wanted to but there would be annoying factors of $180/ \pi$ everywhere. For example, you know that $v= \omega r$ comes from differentiating $s = r \theta$ with respect to time. Had you done that in degrees you'd have an extra factor of $180 / \pi$ and as a consequence you would also have this factor in almost all the equations of rotational dynamics in classical mechanics.

Brain Stroke Patient
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