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Recently, I learnt about physical quantities. When i got to know about plane angle and solid angle, i had a doubt that even though they are just angles, why do we measure it in radians or steradians rather than just degrees?

When i searched about it, some say we need to use that unit in physics and mathematics. Some others say, degrees are part of a 360 degree(which i accept) and radians are the ratio of arc length and radius.

But aren’t they the same? Are they same just like we measure length in m, cm, mm, etc….

Qmechanic
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Steve
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2 Answers2

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For angles, radians are the natural unit, because:

$$ \frac d {d\theta} \cos\theta = -\sin\theta $$

while

$$ \frac d {d\theta}\cos(\frac{\pi}{180^{\circ}}\theta) = -\frac{\pi}{180^{\circ}}\sin(\frac{\pi}{180^{\circ}}\theta)$$

which gets messy fast.

Moreover, there are $2\pi$ radians, and the circumference of the unit circle is $2\pi$.

Likewise for solid angle, the surface area the unit sphere is $4\pi$, which is also the total solid angle.

One can use square degrees, the moon subtends something like $\pi/16$ degrees squared, but that only work out well because it is so small. The whole sky is something like 40,000 degrees-squared, but feel free to calculate the exact value. The exercise should dissuade you from worrying about square degrees.

On geodetic coordinates: I've heard navigators, who did not have electronic computation, liked 360 because its numerous divisors. Note that latitude and longitude are not angles. They can be associated with angles, and the various latitudes (geodetic, parametric, authalic, rectifying, geocentric) are all different...but they are coordinates, and should never be expressed in radians.

JEB
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For planar angles, the need to do radians is coming from the simplicity of the circular trigonometric functions when doing calculus on them. (See JEB's answer.)

For solid angles, there is no tolerable alternative than steradians because while someone might claim that, for single angles, degrees and radians are conceptually equally easy to deal with, in the case of solid angles, the areas are made of $\mathrm d\vartheta$ and $\sin\vartheta\,\mathrm d\varphi$, and there is no simple way to map from the steradian concept to degrees. If it were always $\sin\vartheta\,\mathrm d\vartheta$, we might elect to change from angles to cosines, but it is $\sin\vartheta\,\mathrm d\varphi$, which is a complicated intertwining of two coördinates, and then there is no way to satisfactorily fix this. We are thus left with steradians without competition.

naturallyInconsistent
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