Your explanation of constellation rotation is incorrect. Every constellation rotates completely everyday. So if the Big Dipper is upside down it’ll be right side up in twelve hours. Now if you measure at the same time everyday each constellation will be one degree farther along than at the same time the day before and that is the once a year rotation. Your wikipedia reference has a succint description. “Ancient astronomers noticed that the stars in the sky, which circle the celestial pole every day, seem to advance in that circle by approximately one-360th of a circle, i.e., one degree, each day.”
[…] BetterExplained has a good article on the reasons for using radians rather than degrees. This isn’t something you need to know for A-level, but it should be helpful in understanding why degrees are yanked cruelly away from you when you hit sixth form. […]
[…] No, these critters know their current, instantaneous growth rate, and don’t try to line it up with our boundaries. It’s just like understanding radians vs. degrees”:http://betterexplained.com/articles/intuitive-guide-to-angles-degrees-and-radians/ — radians are “natural” because they are measured from the mover’s viewpoint. […]
Obama needs to add you to the educational advisory board Your method of teaching definitely networks more of the mind enabling better recall and retention. I am sure that when Benjamin Franklin was creating the core of the current educational policies of this nation, this was way closer to the mark of what he intended then the holes in the head that we currently have.
I have a question that I hope you can answer. When talking about a 360 degree circle, 90 degrees is vertical and 180 degrees is horizontal. Why is that different in cardinal directions? Example, 0 degrees is North. I am trying to understand the difference.
@Anon: Thanks :). Yes, I think there are many improvements we can make to how education is handled.
@Kim: Great question. I think the difference is in the starting reference point.
Mathematicians are used to thinking about the x and y axis, so going “right” on the x-axis is the natural starting point for them. Therefore +90 degrees means going “North” (or along the y-axis, as angles increase counter-clockwise for mathematicians).
In navigation (like hiking in the woods), North may be a more universal reference point – it’s where compasses point. In that case, +90 degrees means going East (since the angles increase clockwise).
It’s a bit confusing since each type of use has a different reference point, but thanks for asking.
How do radians and degrees relate to sine space or sines? My understanding of sines is that you simply compute the ratio of the opposite/hypotenuse sides of a right triangle to derive the linear measurement. So can I infer that sines are also from the perspective of the “mover”; since you are dividing one length by another length? If so, would it be better to use sine space because you would have a dimension associated to your unit (cm, m, km, etc…). Maybe a discussion on how the right triangle relates to the unit circle would be helpful. Thanks for a great article!
@E.G.: I’m not quite sure I understand the question, but here’s my take on how sine and radians relate.
Radians and degrees represent progress along a circle; 90 degrees represents a quarter-turn, and pi/2 radians represents the distance traveled when moving a quarter of the way around the circle.
Sine can mean many things, including the ratio of the sides of a right triangle. Another interpretation which may help is that sine represents the “height”, where 1.0 is the max height, -1.0 is the min height, and everything else is a fraction in-between.
(Edit: correcting an error in this comment):
The interesting thing is that sine/cosine represent position in grid coordinates, which the mover may not know about!
For example, 45 degrees represents a certain position along the circle. From the mover’s perspective, they are halfway to 90 (top of the circle), and indeed, they have moved halfway to their goal (at 45 degrees, the distance along the circumference from the start and top of the circle is the same).
But from our observer’s perspective, 45 degrees looks like a height of sin(45) or .707 – that is, at 45 degrees, the mover is 70.7% of the way to the top! In the last “half” they move the remaining amount. I see sine and cosine as ways for us to map the distance traveled in the mover’s frame of reference to distance traveled in ours.
Radians and degrees are different ways of describing how far you’ve traveled along the circle. Sine is a way to describing how ‘high’ you are on the circle (from our grid’s perspective), as a percentage of the maximum. Hope this helps!
[…] There are, let’s face it, any number of websites out there claiming to offer help to pupils who are having difficulty with maths, and you’d have to say the success rate of these sites is somewhat limited. But to my mind this particular website is a real success, because it goes one better: it offers better explanations for teachers. I’ve had a quick rummage round the website and am already impressed at how the author manages to give fresh insights into such matters, as, for example, this beauty on why radians are so useful. (Yes, I know, you can differentiate with radians… and degrees are arbitrary… but there’s even more to it than that, I assure you.) I’m not saying you’ll agree with everything he says, but the site will make you think. […]
I finally got this radian stuff and thought I would sum up my brain blast for anyone who is still confused. This is a really basic explanation that just uses straight math.
Describing an angle in radians is just a way of writing an angle without the degree symbol.
The measure of an angle in radians is the ratio of the arc length it cuts out to the length of the circle’s radius.
If the arc that the angle cuts out is exactly equal to the length of the radius, the angle therefore has a measure of 1/1, 1 radian, or just 1.
Another useful example:
If the angle cuts out an arc that is equal to the whole circumference of the circle (2πr), the normal angle is 360 degrees, and so that means that the ratio of the arc length (2πr) to the radius is just 2πr/r, or 2π. This gives us the measure of the angle in radians, 2π.
In other words, 360 degrees = 2π radians,
180 degrees = π radians, and
180 degrees/π radians = 1.
We can use this ratio as a conversion factor.
To convert 39 degrees to radians, multiply 39 degrees x (π radians/180 degrees) to find that 39 degrees is really 39π/180 or about 0.68 radians!
Hi Kalid I’m in grade 11 doing a college course supplied by our school and right now I think radians are the most stupid “number” in the world. Thanks for the explanation on what the stupid things are any way at least I’ll be able to understand part of it.
Your method of teaching definitely networks more of the mind enabling better recall and retention. I am sure that when Benjamin Franklin was creating the core of the current educational policies of this nation, this was way closer to the mark of what he intended then the holes in the head that we currently have.