Radians vs Degrees: When to Use in Calculus & Physics

[Llama77]

So I have a few different calculators I use. I am currently in a Calculus 1 and Physics 1 course. Both for engineers.

As am example of what I am trying to explain, such as my physics HW It asked me to calculate the magnitude of the Vectors A+B+C. I do so and It worked fine, I did this is radians, But then it asked me for the degrees of the added vectors and I gave it the answer but it said i was wrong. Now my calculation were right, all except I think i should have been in degree's rather than radians. When i redid all the math in degrees the degrees section was now correct.

So I am looking for some advice on knowing which to use, as I have been told by my calc professors that Degree's is Barbaric or something to that nature.


Thank you.

 

[arildno]

What's barbaric about a quantitiy proportional to radians??
Poor degrees! I'll rush to their defense any day..

 

[Bitter]

You use several different calculators? T-83 and beyond can switch from degrees to radians.

Anyway, the best way to know which to use is by context. If the variables in the problem deal with degrees, use degrees, if they are just constants, use radians.

 

[mathwonk]

radians are clearly the best abstractly, but degrees are the most understandable. i.e. everyone in my generation knows what 90 degrees means, and we do not all know quickly what pi/2 radians means, much less 2 radians.

measuring angles in degrees is sort of like meaasuring lengths in cubits.

 

[Llama77]

so if the problem is going to ask for degrees or had degrees in it then I should just stick with degrees for ?

 

[arildno]

Indeed you should!

You should always answer the question, whatever the question is.

 

[MeJennifer]

radians are clearly the best abstractly

Well I know how it is derived but still it is not clear to me at all that it is "clearly the best".
Especially not abstractly!

In abstract terms I would actually prefer: 1c \iff 360^\circ \iff 2\pi rad,
where c stands for circumnavigation.

Anyway it is just a convention and radians seems to be the clear winner.

 

[arunma]

So I have a few different calculators I use. I am currently in a Calculus 1 and Physics 1 course. Both for engineers.

As am example of what I am trying to explain, such as my physics HW It asked me to calculate the magnitude of the Vectors A+B+C. I do so and It worked fine, I did this is radians, But then it asked me for the degrees of the added vectors and I gave it the answer but it said i was wrong. Now my calculation were right, all except I think i should have been in degree's rather than radians. When i redid all the math in degrees the degrees section was now correct.

So I am looking for some advice on knowing which to use, as I have been told by my calc professors that Degree's is Barbaric or something to that nature.


Thank you.

Are you sure that you entered in the right quantity when you were doing your trig calculations. For example, if you put a number into a trig function, and your calculator thinks it's operating in radians when you were using degrees, then you'll obviously get a wrong answer.

Anyway, to answer your question, I think you'll find that both degrees and radians are used in physics. It really depends on context. For example, when talking about the phase of an AC signal or the angular displacement of a rotating object, we'll typically use radians. When we talk about the angle at which a projectile is launched, degrees usually win out. To put it simply, degrees are easier to visualize, but radians are more elegant and easier to work with mathematically.

Whatever you do, always use radians when you're taking a derivative of a trig function!

 

[HallsofIvy]

The functions used in mathematics, as opposed to calculations in, say, physics have no "units". If f(t)= t², neither the t nor f(t) are measured in feet or meters- they are just numbers.
Similarly, in the functions sin(t) and cos(t), x has no units- in particular they are not measured in degrees or radians.

In fact, one definition of sign and cosine widely used in calculus or pre-calculus measures the variable t around the circumference of a unit circle. It is not an angle and so cannot be measured in degrees or radians.

Of course, calculators are designed by engineers, not mathematicians, and they think of sine and cosine in terms of angles (look at the "phase angle" in waves where there are no angles at all!). The radian measure of an angle is really the distance around the circumference of a unit circle subtended by that angle and so corresponds to the "circle" definition. The "x" in sin(x) or cos(x) corresponds to radian measure of an angle. That is, I am sure, what your math professors were telling you (did they really use the word "barbaric"??).

In your vector problem, you are doing an application of mathematics in which there really are angles so that either radians or degrees (or even "grads"- one hundredth of a right angle) would be appropriate. If "then it asked me for the degrees of the added vectors", then it would be foolish of you to use anything other than degrees!

 

[knowNothing23]

I'm still unsure about, when to use radians or degress in my calculator.

Could anyone give an example of both cases?
What should I use, if I'm to find sin(2)?
Why is that sometimes the difference in radians or degrees do not matter?

Thank you.

 

[symbolipoint]

I'm still unsure about, when to use radians or degress in my calculator.

Could anyone give an example of both cases?
What should I use, if I'm to find sin(2)?
Why is that sometimes the difference in radians or degrees do not matter?

Thank you.

What does "2" measure? Use the unit in your calculator process. If the "2" measures degrees, then set your calculator for degrees. If the "2" measures radians, then set the calculator for radians. If you do not want to reset your calculator between degrees and radians, then simply use the ratio of 360 degrees equals 2∏ radians.

 

[knowNothing23]

What if there's no indication of what's required? Neither radians or degrees?

 

[Integral]

Do the problem in the angle measure of your choise. Give the answer in both.

 

[symbolipoint]

What if there's no indication of what's required? Neither radians or degrees?

Then the question was not written right and is meaningless, unless you know in advance which unit is intended.

Do the problem in the angle measure of your choise. Give the answer in both.

That's one approach, but the problem description needs to be sufficiently given.

 

[HallsofIvy]

As I said in my previous post, when a trig function appears purely as a "function", without reference to an angle, then the units should be "radians" (strictly speaking it has no units but to put it into a calculator, you must use radians).

I agree with symbolipoint that, if a problem deals with angles, then the problem should state whether it is measured in degrees or radians. But if you simply have sine or cosine functions, without reference to angles, you can assume radians are intended.