Degrees versus Radians

[mmm4444bot]

The following quote comes from another thread. The discussion below was moved from that thread to this one.

I would draw a picture, then don't worry about the sign of -2/5 too much

I'm curious about the meaning of your phrases "don't worry" and "too much".

The sign of tangent's inverse-function input is crucial here.

:???:

 

[Yogi]

In rectangular coordinates you start at the origin and then move x units left or right and then y units up or down and end at the point you want. In polar you start at the origin, move r units along the positive x axis, then rotate this point by an angle of theta. Think of it like a compass, you put the sharp end at the origin, the pencil on the positive x axis r away from the origin, and rotate the compass theta degrees or radians. Example: (1, 90 degrees) in polar is (0,1) rectangular.

It is looking for an answer of the form (r,theta) so let's use radians for theta since your formula uses Pi. Draw the x and y axis, draw the point (-5,2), draw a right triangle with vertices (0,0) (-5,0) (-5,2). From this we know the angle closest to the origin is .38 radians(calculator, or trig table). Start at 0 radians, rotate to Pi radians(180 degrees), then up into the second quadrant by .38 radians is the same as rotating +Pi-.38 radians = 2.76 radians. When rotating, positive is counter clockwise negative is clockwise. So the polar coordinates are (√29, 2.76 radians). I would draw it out like this a few times before trusting the formula completely.

Using degrees isn't necessarily a mistake, his Pi in the original post looked like an n and polar coordinates can use either method. If his problem has a formula with Pi in it then yes use radians. Degrees are measured in numbers as well btw -_-

 

[Yogi]

I'm curious about the meaning of your phrases "don't worry" and "too much".

The sign of tangent's inverse-function input is crucial here.

:???:

It's only crucial if you actually use that formula which is pretty useless. The lengths of a right triangle are positive. The angle inside is positive. For the example given rotate +(counterclockwise) Pi radians and -(clockwise) angle to find theta in polar. Remembering 0 formulas is much easier than remembering 2 different formulas, each of which only work in 2 specific quadrants.

 

[pka]

Using degrees isn't necessarily a mistake, his Pi in the original post looked like an n and polar coordinates can use either method. If his problem has a formula with Pi in it then yes use radians. Degrees are measured in numbers as well btw -_-

You are simply wrong about that.
In current mathematical educational practices degrees are simply not usred.
As I said, the degree mode is not used as a default setting for any major calculator or CAS.
It is just not mathematical.

 

[mmm4444bot]

[The sign of arctangent's input is] only crucial if you actually use that formula

[Using those formulas are] pretty useless.

Those formulas have many valid applications, the very least of which is educational. After all, the original poster's course is using them. Hence, the sign on -2/5 is crucial, in the context of [the original thread].

And, not every person using polar coordinate systems outside of academia resorts to drawing triangles when they need to do a conversion. Just sayin'.

If you would like to start a thread to discuss the merits of any particular formula, we enjoy reading that stuff on the Math Odds & Ends board. Cheers :cool:

 

[Yogi]

@pka Define current mathematical educational practices, do you mean college level? I know in high school in the U.S. we don't learn about the elusive .5236-1.0472-1.5708 radian triangle, we learn 30-60-90. Also, degrees are just easier for high school kids to get a grasp of. How are they not mathematical? There is a simple mathematical conversion from one to the other. Just because you don't use them after high school doesn't mean they don't exist. Guess what, you learn tons of stuff in high school that you never use.

Regardless, from a learning standpoint, drawing these out a few times is miles ahead from simply plugging some numbers into a formula and writing down the answer.

 

[mmm4444bot]

I know in high school in the U.S. we don't learn about the elusive .5236-1.0472-1.5708 radian triangle, we learn 30-60-90.

I do not claim to know what high schools are doing these days, but I was introduced to the 30-60-90 triangle in the 8th grade.

degrees are just easier for high school kids to get a grasp of.

What high school kids?

Are you advocating for all US high schools to make things easier for their students?

By the way, did you happen to see or hear anything about a recent, national survey in the news which reports that high school students in the US are -- in general -- bored with what they perceive as "non-challenging" math courses?

There is a simple mathematical conversion from one to the other.

Exactly. Simple enough even for high school students who may need something easy to grasp.

Just because [a person does not] use [degree measurement] after high school doesn't mean [degree measurement does not] exist.

Well, of course. Carpenters use degree measure almost exclusively.

Both units of angle measure (degrees and radians) are valid, as are other established units for measuring angles.

You may argue the merits of what high schools teach, but you may not advocate for the invalidation of any established system of measurement by simply arguing that it's not your personal preference.

There's more than one way to skin a cat, in other words. And, perhaps, I give high school students (in general) more credit than you do.

Cheers :cool:

 

[mmm4444bot]

from a learning standpoint, drawing [triangles] a few times is miles ahead from simply plugging some numbers into a formula

From a learning standpoint, perhaps students should do what their teachers instruct first.

I would expect that students will have already drawn a few triangles, before they're exposed to polar coordinate systems.

Again, I'm not sure what high schools do in general, anymore; yet, I am confident that as students proceed through and from high school, they will be exposed to several different ways of thinking about certain concepts.

There is more than one way to skin a cat. :cool:

 

[Yogi]

Why does this thread exist again?

You may argue the merits of what high schools teach, but you may not advocate for the invalidation of any established system of measurement by simply arguing that it's not your personal preference.
Cheers :cool:

I don't recall advocating anything, if you read the posts at all I originally did the problems in degrees, and then later in radians. My only issue was pka using words like "mistake" in reference to using degrees when in the original problem this was not clear, saying degrees aren't measured in numbers(false), and they are never used(also false). I don't think I ever said I only use degrees or we should get rid of radians all together or any such nonsense. Yes, I am well aware that most calculus level polar problems are done in radians. Did you actually read the posts or just one while ignoring everything said before it? The rest of your wall-of-text is kind of off-topic and feels like arguing(poorly) for the sake of hearing yourself argue.

From a learning standpoint, perhaps students should do what their teachers instruct first.

It sounds like he did, then came to these forums...perhaps the instructions to plug numbers into a formula weren't good enough. Then I suggested to draw it to understand what is going on. What exactly is the problem here?

 

[mmm4444bot]

Why does this thread exist again?

This thread exists so that you may express your opinions about "useless formulas" outside of somebody else's discussion.

Then I suggested to draw

What exactly is the problem here?

You also implied that the original poster should ignore instruction methods received in the classroom. (By the way, the original posting is clear regarding whether radian measurement should be used because the formulas given in the classroom are explicity stated. Somebody who shuns radians might not notice the connection.)

Well, if there is a "problem", then perhaps it has something to do with your choice of words. I'm not sure. Are you able to expound on your perception of "problem"?

Otherwise, this thread is just another discussion. Cheers :cool:

 

[Yogi]

As I said before and you probably didn't notice since you clearly haven't read the posts, in the original posters formula Θ=π+tan^-1(2/-5) the pi looks like an n, in which case degrees or radians could work. In fact the original answer he got of 18.66 was much closer to the actual answer in degrees. It wasn't until la later post where he actually types "Pi" that it becomes 100% clear that radians should be used.

I never shunned radians, that's just something you fabricated in your mind.

After you understand it doing it like that a couple times then go to your formula

The original problem in his first post asked to translate from rectangular to polar. It says nothing about the use of the formula being mandatory. What are these "methods" I am having him ignore? The method of mindlessly plugging numbers into an equation without knowing why? After you understand, use it, don't use it, I don't really care. I clearly say "go to your formula", perhaps you should work on your reading comprehension a bit.

I call the formulas useless because it isn't very efficient to commit them to memory, if you don't have them on hand it's a pain to look them up, they have limited use(you have to know x and y), there are 2 different ones that only work in 2 quadrants each, and all the formula really does is add two angles together. Do you really need a formula to add two angles together? I don't, maybe that's why I can do this problem 11 years after learning it(with no practice since then) without looking anything up. It's like saying z equals the sum of 8 and -2, oh btw here's this cool formula you can use f you need it(z=x+y). -_-

 

[mmm4444bot]

You are one passionate individual, Yogi. I like to see that!

 

[kenrapp1]

Degree to Radian Conversion.. please solve... disaggreement in our home on the answer

225 degrees..convert to Radians. My answer... 225 Degrees x Pi/180 = 225/180 x Pi equals 5Pi/4.

Other answer was 225 x Pi /180 = Pi/1.25. If you don't mind, please show the stepe so we can resolve our disagreement.

 

[HallsofIvy]

225 degrees..convert to Radians. My answer... 225 Degrees x Pi/180 = 225/180 x Pi equals 5Pi/4.

Other answer was 225 x Pi /180 = Pi/1.25. If you don't mind, please show the stepe so we can resolve our disagreement.

That "other answer" is simply not correct. 225/180= 45/36= 5/4= 1.25. The answer "5Pi/4" is equal to Pi times 1.25, not divided by it.

 

[lookagain]

225 degrees..convert to Radians. My answer... 225 Degrees x Pi/180 < < = 225/180 x Pi equals 5Pi/4.

This is incorrect. You must have the units in each appropriate place.


(225 degrees)*[(pi radians)/(180 degrees)] = (5pi/4) radians = 1.25pi radians


\(\displaystyle Or, \ \ \dfrac{225 \ degrees}{1}\bigg(\dfrac{\pi \ radians}{180 \ degrees}\bigg) \ = \ \dfrac{5\pi \ radians}{4} \ = \ 1.25\pi \ radians.\)