1. Jul 28, 2013

### vee6

How to convert the -300 deg (minus 300 degree) to a radian?

2. Jul 28, 2013

### rock.freak667

π rad is equivalent to 180 deg.

so 1 deg = π/180 rad

(π =pi if it does not come out too clearly)

3. Jul 30, 2013

### vee6

-300 deg = 60 deg

am I correct?

if so

is it correct?

4. Jul 30, 2013

### Mandelbroth

A unit vector directed at -300 degrees is coterminal with a unit vector directed at 60 degrees, but the angles are not equal. You are not correct.

5. Jul 31, 2013

### HallsofIvy

What Mandelbroth is saying is that -300 is NOT "equal" to 60 but that since 360- 300= 60, they are equivalent and have the same trig function values.

6. Jul 31, 2013

### vee6

So

Is it correct?

7. Jul 31, 2013

Yes.

8. Aug 1, 2013

### HallsofIvy

Yes. Also note that since, as above, -300 degrees is equivalent to 60 degrees, we have the -300 degrees is equivalent to 60(pi/180)= pi/3 radians (If you are using "*" to indicate multiplication, do NOT write "* rad"). This is true because one/3 complete revolution is 2pi radians and 2pi- 5/3 pi= (6/3- 5/3)pi= pi/3 so that -(5/3)pi radian is equivalent to pi/3 radians.

9. Aug 1, 2013

### Staff: Mentor

Edited to remove a typo.

10. Aug 1, 2013

### SteveL27

I disagree with this statement. If I see a vector making an angle of 60 degrees with the positive x-axis, how would I know if it "got there" by rotating 60 degrees counterclockwise or 300 degrees clockwise? Am I expected to examine its travel itinerary? Of course this is nonsense.

The two angles are equal, not "equivalent but not equal." You might as well argue that in the integers mod 5 the classes [1] and [6] are "equivalent but not equal." If an algebra student said that we would correct him, not encourage his misunderstanding.

It's true that the integers 1 and 6 are equivalent mod 5 but not equal. But the elements [1] and [6] in the integers mod five are equal, not "equivalent but not equal." When we're discussing angles, we are living in the reals mod 360, in which -300 = 60, or [-300] = [60] if we are being notationally pedantic.

Since this is an elementary question, I would not want to leave readers confused on this point. The angles 60 degrees and -300 degrees are equal angles, end of story. When we encounter an angle in the plane, we are not expected to examine its travel history to see how it got there. There is no way to distinguish an angle of -300 degrees to one of 60 degrees.

Last edited: Aug 1, 2013
11. Aug 1, 2013

### jbunniii

They are equivalent as angles, but they certainly are NOT equal. Try expanding sine or cosine in a Taylor series using only finitely many terms, and compare the results if you plug in -300 degrees vs. 60 degrees.

12. Aug 1, 2013

### SteveL27

I didn't believe they would be different so I wrote a little program to test it out. If you plug pi/3 and -5pi/3 into the Taylor series for cosine, they start out wildly different for a few terms but both settle down to 1/2, good to 6 decimal places, after about 12 terms. That's not bad at all, and it contradicts what you claim.

If I handed you a piece of paper showing an angle of 60 degrees; and another piece of paper showing an angle of -300 degrees; how would you tell them apart? If they were on semi-transparent paper, you could overlay one on top of the other and they'd be identical. They are two names for the same angle. You can only regard them as different if you consider the paths a point would travel in tracing out the angles; but that's not really part of the definition of an angle. I suppose this is all a distinction without a difference, but still ... I'd like to know how you would tell them apart.

Last edited: Aug 1, 2013
13. Aug 1, 2013

### jbunniii

They won't be exactly equal with only finitely many terms.

Theoretically they are indistinguishable if you consider the points on the unit circle to which they correspond. But what about other curves? If I have a closed curve which has a winding number greater than 1, I can certainly distinguish between -300 and 60. See the animated example here: http://en.wikipedia.org/wiki/Winding_number

Indeed, the very definition of winding number requires that we distinguish between angles that differ by a nonzero multiple of $2\pi$ radians:
$$\text{winding number } = \frac{\theta(1) - \theta(0)}{2\pi}$$
where $\theta(0)$ and $\theta(1)$ are the initial and final angles at which the curve starts to repeat itself.

14. Aug 1, 2013

### SteveL27

So what? Neither truncated sum is equal to the correct answer after finitely many terms. But the convergence pretty is fast for both of them. As I said they're both within 6 decimal places of 1/2 (the correct answer) within a dozen terms. What does it matter that the truncated sums differ? You wouldn't expect otherwise.

Winding numbers have nothing at all to do with this. If I show you an angle of 60 degrees, it's the same as an angle of -300 degrees. You are introducing conceptual elements to the situation that aren't there in the definition of an angle. I agree that if someone ran around a circle and returned to their starting point, that would not be the same as staying in place. They would have expended energy, etc. But the angle would be zero in either case, assuming they started from (1,0) and ran around the unit circle.

The analogy I gave earlier is the distiction between saying:

a) The integers 1 and 6 are equivalent, mod 5, but they are not the same number.

b) The elements [1] and [5] of the integers mod 5 are equal.

If you're striving for conceptual precision, it's helpful to distinguish between an angle, on the one hand; and a point travelling around a circle on the other. The concept of "travel" is not part of the definition of an angle. It's something you're adding in to obfuscate the definition of an angle.

Last edited: Aug 1, 2013
15. Aug 1, 2013

### jbunniii

For another example where the distinction is important, consider the oft-stated notion that the frequency of a sinusoid (say, varying as a function of time) is the derivative of its phase:
$$f(t) = \frac{d}{d\theta}\phi(t)$$
and that the phase can be obtained by integrating the frequency:
$$\phi(t) = \int f(t) dt$$
These relations show up in many aspects of physics and engineering, for example in FM radio a sinusoidal carrier is used, and the phase is varied in proportion to the amplitude of the message being transmitted.

If we insist on "wrapping" $\phi(t)$ modulo $2\pi$, then these statements are false for any function $\phi(t)$ which crosses zero: such a function is not even continuous, let alone differentiable, at points where this occurs. This is a rather arbitrary and artificial problem which can easily be fixed by maintaining the distinction between phases that differ by a multiple of $2\pi$.

16. Aug 1, 2013

### jbunniii

I guess I don't see what your complaint was with HallsOfIvy's statement:
He is saying exactly the same thing as you are. -300 and 60 are not equal, but [-300] and [60] are, if we define the equivalence classes to indicate angles which correspond to the same point on the unit circle.

17. Aug 1, 2013

### Simon Bridge

Sure - travelling through 60 degrees of arc will put you in the same place as travelling -300 degrees of arc ... but how you get there may be important: i.e. there may be a wall blocking one direction. Even if you just rotated on the spot, one way will take longer and/or require more torque than the other way.
The process can matter.

So what was the question again?

18. Aug 1, 2013

### SteveL27

At the risk of flogging this deceased equine into the ground, an angle of -300 degrees is equal to an angle of 60 degrees. That is my statement. I objected to what HOI said because he's being imprecise; and because a lot of students read this forum; and because degrees and radians are often confusing to students; and conceptual clarity is a virtue.

Here's another example. I have two pieces of paper, each with the numeral '3' on them. You say they are equal. I say they are not equal, they're only "equivalent," and I offer the following admittedly farfetched but mathematically correct argument.

The first piece of paper has a representation of the number 3 on it, but I got that by starting with the real number 3 and writing down its English-language decimal numeral representation.

The second piece of paper also has a representation of the number 3 on it, but I say I got that by starting from the complex number 3 + 5i and taking the real part. Then I say, "So you see, 3 and 3 are equivalent modulo the relation on the complex numbers of having the same real part; but they are not really equal."

You would NEVER claim that Re(3) and Re(3 + 5i) are merely "equivalent but not equal." That's simply wrong. They are equal as real numbers; even though 3 and 3 + 5i are merely equivalent mod having the same real part.

And the point is that when you see the numeral '3', you are not expected to have secret off-the-books information that we got that 3 by taking the real part of 3 + 5i. You're given 3, that's what it is. There's no additional secret information stuck to the bottom or hidden inside.

But that's exactly what you and HOI are claiming: That two absolutely identical angles are not actually equal, but only equivalent, because you have secret information of how they were constructed. I would most sincerely ask you: Given the angle on a piece of paper, how would I know the angle's construction history? Given the number 3, how would I know that you secretly regard it as the real part of 3 + 5i, to be considered "equivalent" but not equal to the real part of 3 + 0i?

Denying the equality of the angles of -300 and 60 is an obfuscation of the definition of angle that, when repeated by gold handles, does not serve students trying to understand this already-confusing material precisely.

I don't think I have anything to add so I'm done here, since I've just been saying the same thing repeatedly. Clearly some people disagree. I think that if people would consider the actual definition of an angle, they would change their minds.

Last edited: Aug 1, 2013
19. Aug 1, 2013

### Integral

Staff Emeritus
The OP has not been involved in this thread since post#7 and it has drifted way off topic. Enough already