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Degrees seem to be so much easier to work with and more useful than something with a $\pi $ in it. If I say $33\:^\circ $ everyone will be able to immediately approximate the angle, because its easy to visualize 30 degrees from a right angle(split right angle into three equal parts), but if I tell someone $\frac{\pi }{6}$ radians or .5236 radians...I am pretty sure only math majors will tell you how much the angle will approximately be.

Note: When I say approximately be, I mean draw two lines connected by that angle without using a protractor.

Speaking of protractor. If someone were to measure an angle with a protractor, they would use degrees; I haven't seen a protractor with radians because it doesn't seem intuitive.

So my question is what are the advantages of using degrees? It seems highly counterproductive? I am sure there are advantages to it, so I would love to hear some.

PS: I am taking College Calc 1 and its the first time I have been introduced to radians. All of high school I simply used degrees.

James Smith
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  • Have you gotten to derivatives of the trig functions yet? – Tim Raczkowski Dec 03 '15 at 00:20
  • The main nice property is that an arc of an angle $\theta$, measured in radians, has length $r \theta$, where $r$ is the radius of the circle. Similarly the area of the sector is $\frac{1}{2} \theta r^2$. That first property makes the trig functions much, much nicer in calculus, ultimately because you have $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$. In degrees you get $\frac{\pi}{180}$ instead. – Ian Dec 03 '15 at 00:27
  • Let me ask you this: What is arc length of of an angle 42.87 degrees along a unit circle? What is the area enclosed by it? If I give the same angle through radians to you, the answer to these questions is immediate. – Hamed Dec 03 '15 at 00:28
  • Related: Why do we require radians in calculus? – Andrew D. Hwang Dec 03 '15 at 00:35
  • $\pi$ is just halfway around the circle. If you want $\pi/6$, you go one sixth of the way around that protractor. The mistake may be trying to think of $\pi/6$ as some decimal number $0.52\ldots$ instead of just keeping it as "one sixth of the way to the other side". If you think that way, it's just as easy for anyone to sketch that angle. – 2'5 9'2 Apr 16 '24 at 04:26

3 Answers3

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Radians are a dimensionless measure; being initially defined as the arc length of a circle circumscribed by the angle divided by the radius of that circle.

This comes into play when dealing with derivatives of trigonometric functions.

$$\dfrac{\mathrm d \sin(x)}{\mathrm d x} = \cos (x)$$

versus:

$$\dfrac{\mathrm d \sin (x^\circ)}{\mathrm d x} = \dfrac{\pi \cos(x^\circ)}{180^\circ}$$

It's a lot more convenient to use radian measures in calculus.

Graham Kemp
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  • Degrees are a human-imposed unit. It was once believed a year had 360 days. Hey, the number is divisible by 1,2,3,4,5,6,8,9 and 10. – ncmathsadist Dec 03 '15 at 01:12
  • Now cut me a slice of $\pi$. – ncmathsadist Dec 03 '15 at 01:12
  • @ncmathsadist Well by that argument all of mathematics has been imposed by humans. Something as basic as our number system was an arbitrary tool used to compare quantities to some standard values we invented. Before people would count by matching pairs, but the human-imposed system has proved to be valuable. – James Smith Dec 03 '15 at 21:13
  • Also this is a nice answer. I guess deriving values is more complex with degrees and radians is used in calc purely for simplicity. I still think degrees are simpler in everyday scenarios, but for calc I am glad we have radians now – James Smith Dec 03 '15 at 21:19
  • The right hand side of your example should be $180\cos(x^{\circ})/\pi $. But you can get a similar equation by differentiate $\sin(x)$ with respect to $x^{\circ}$. So the example does not show the inconvenience of using degrees. – chichi Apr 16 '24 at 03:51
  • @chichi In $\sin(x^\circ)$, $x^\circ = \frac{\pi}{180}x$, where $x$ is the number of degrees in an angle, which we convert to radians in order to apply the standard $\sin(\cdot)$ function. If you are computing all your sines using degrees, it makes perfect sense to differentiate this with respect to number of degrees. But if $x$ is the number of radians in an angle, then $x^\circ$ is the number of $(180/\pi)$-radian units in the angle. It measures angles in a unit that makes no intuitive sense, so there's no reason we should want to differentiate with respect to it. – David K Apr 16 '24 at 17:34
  • @Graham The only error in your original answer was that you had the degree symbol in the denominator on the right-hand side. The coefficient is simply $\pi/180$. The comment about $180\cos(x^{\circ})/\pi$ has things backward. – David K Apr 16 '24 at 17:38
  • @DavidK I think $x^{\circ} = x * 180/\pi $, where $x$ is measured in radians and $x^{\circ}$ is measured in degrees. The chain rule takes care of everything so $d\sin(x^{\circ})/dx^{\circ} = \cos(x^{\circ})$ with no additional factors. There are surely reasons why the use of radians is more natural but it is not shown in differentiation formulas like this. – chichi Apr 17 '24 at 02:34
  • @chichi You can think whatever you like, but the ISO defines the degree symbol ($^\circ$} as the numerical factor $\pi/180$, not $180/\pi$. Moreover, if you plot $\sin(x^\circ)$ from $x=0$ to $x=90$, ($\sin(0^\circ)$ to $\sin(90^\circ)$), you'll see that it takes $90$ units of "run" in order to get just $1$ unit of "rise", and the slope is everywhere much smaller than $1$. The maximum slope is $\pi/180$. But if instead of treating $\sin(x^\circ)$ as a function of $x$ you let $u=x^\circ$ and differentiate with respect to $u$, then you're just doing everything in radians. – David K Apr 17 '24 at 04:10
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The radian is the standard unit of angular measure, and is often used in many areas of mathematics. Recall that $C = 2\pi r$. If we let $r =1$, then we get $C = 2\pi$. We are essentially expressing the angle in terms of the length of a corresponding arc of a unit circle, instead of arbitrarily dividing it into $360$ degrees.

The reason that you believe degrees to be a more intuitive way of expressing the measure of an angle, is simply because this has been the way you have been exposed to them up until this point in your education. In calculus and other branches of mathematics aside from geometry, angles are universally measured in radians. This is because radians have a mathematical "naturalness" that leads to a more elegant formulation of a number of important results. Trigonometric functions for instance, are simple and elegant when expressed in radians.

For more information: Advantages of Measuring in Radians, Degrees vs. Radians

nicole
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If we ever come across an alien civilization there is one thing you can rely on: Their mathematicians will use radians and not degrees, simply because lots of formulas and equations work with radians only (which is why mathematicians consider radians to be natural while degrees are totally arbitrary), like for example the Taylor series for sine and cosine or Euler's identity. If you use degrees instead of radians you would have to divide them by 180º and multiply them with π, which would result in elegant formulas and equations becoming a lot less elegant. The reason you prefer degrees is simply because the concept of radians is new to you. You will get used to them in time.

And honestly: Saying that the sum of angles inside a triangle equals 180º is rather dull, but saying that it equals π looks exciting and like a profound insight.

BaldFriede
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