do the angles 60 and -300 have the same magnitude?

it probably sounds silly but I am curious, do they have the same magnitude?

Also when talking about direction angles, do the angles 60 and -60 have the same magnitude?

What composes an angle then? How can you tell that two angles are the same? Do they have to have the same MEASURE as well as DIRECTION to be considered the same?

Thank you in advance! :)

Re: do the angles 60 and -300 have the same magnitude?

You will have to explain what **you** mean by "magnitude" of an angle! One common use of "magnitude" of a **number** is "absolute value" but, in that sense, "-300" and "60" definitely do NOT have the same magnitude so apparently that is not what you are. It is, of course, true that 60= 360- 300 so that angles of "-300 **degrees**" and "60 **degrees**" give the same point on the unit circle so that sin(-300)= sin(60) and cos(-300)= cos(60). But that is a rather limited use of angles. I would say that is a property of sine and cosine (they have period 360 degrees) rather than an inherent property of the angles themselves.

Re: do the angles 60 and -300 have the same magnitude?

Agree with HallsofIvy. One more thought though, is that angles like -300^{o} and 60^{o} are called "coterminal" angles. This means that if one side of the angles is set (if they both start from the same point), then they will end up at the same point.

And, as with HallsofIvy's response to your inquiry about "magnitude", the response to your question about "the same" depends on your meaning. If you mean "congruent" (rigid transformations can move one onto the other, like you could use in analyzing polygons or polyhedra), then 60^{o} and -60^{o} are the same. In fact, most of those types of applications do not even include the sign of an angle as an indication of direction. But if you mean "coterminal" (like you could use in a vector or navigation problem), then these two angles are not the same.