@tin: You’re welcome!
[…] trig: 'x' is radians (they are more natural!), and a full cycle is going around the unit circle (2*pi […]
Show me a sextant with radian reading and I accept that radians have place also elsewhere than in the nerds’ world. (Or a GPS receiver. Or a compass. Or a milling machine turntable. Or just any practical implementation.)
I mean, radians are useless in the real life. There are absolutely no practical applications for them, so why bother? Why are the longitudes and latitudes on the map measured in degrees and not in radians? When a sailor or pilot takes a bearing, does he use degrees or radians?
Nerds may be clever, but they usually fail in practical implementations.
@Ironmistress:
The article lists some examples of use. For example, if someone says the tire spun 2000 degrees per second, it takes a lot more work mathmatically to find out just how much it’s really spinning, whereas radians are as easy as multiplying it by the radius.
Mike, not any more than making calculations in nautical units than in metric. (The usual units for spinning is revolutions per minute, not radians per second as the SI system would imply).
That is the reason why the Nautical Almanac lists the various celestial quantities in degrees, minutes and tenths. We are used to them. That is why we use nautical miles instead of kilometres while at sea. One nautical mile is the same as one angular minute of latitude.
@Ironmistress: Great question. I think the main issue is realizing there are several viewpoints we can take.
When we see an orbiting satellite, do we measure its speed by how fast it goes by our field of view (degrees/sec) or how fast it’s going around its cycle? (radians/sec). When you’re driving around the highway at 60mph, you think that’s your speed… but from the perspective of a gnome at the center of the earth, you are moving .00001 degrees per hour. You care about mph; the gnome cares about degrees per hour.
In most nautical applications, we’re the “gnome” and want things from our own perspective. That’s fine.
But the equations of physics/astronomy that enabled the creation of the sextant, etc. are best described with equations in radians (sine, cosine, pendulum motion, etc. are described in radians, http://en.wikipedia.org/wiki/Sun-synchronous_orbit).
Utility depends on context – the military uses “mils” (1/1000 of a radian) for firing tables, etc. and not degrees. Hope this helps!
Thank you so!!!
First time I saw you blogs last week. And since last week I kept on reading various articles written by you (adding 1 to 100, birthday paradox, e, natural log etc). I enjoyed all the them. Your ideas and understanding of mathematics is crystal clear!! Wow!!. It helped me to understand mathematics quite natural way.
Thanks for all your post…
Please provide some insight into steradian (solid angles). I would like to know how total angle in sphere is 4*pi.
@Quaid: Thanks for the note, glad you’re enjoying the site! Great suggestion, I’ll add it to my list :).
Hey Kalid,
Great article, but I still have a intuitive gap on sin(x)/x
If we look at the plot
You explained the first peak, at x=0, but how do you explain the other peaks at x~=8?
My intuition says that peak represents when we’ve made a full circle and have just barely passed the x-axis again. Is that right?
Big fan of yours Kalid, please keep these tutorials coming. Your website along khan’s academy are the best resources for understanding math.
I wish my high-school math teacher explained as you and that I paid more attention.
Thanks!
I dropped out from architecture when I was a teenager because the Calculus lectures were so utterly cryptic -and no teacher I knew would make it any clearer- that this was a source if horrible stress I simply could not cope with.
Now, at 40, I am considering taking back my original wish of being an architect, and by having seen your explanations -addresses for right brain people like myself, I now have hope.
THANK YOU SO MUCH!!
Your writing is hilarious, simple, and so easy to understand. Other teachers (especially math) should learn your style. I love your examples, too. Thank you and keep up the excellent work.
Sam
I was just about to congratulate you on the clarity of your explanation and to finally help me make sense of the difference between degrees and radians, but before I do, I guess I should make sure that I got this right: if I understand correctly, one way to put it would be that a degree is in fact an angular measure where a radian is more like a distance unit? Also, 45 degrees will always be 1/8 of a complete circle and nothing wil ever change that, but the radian value will vary based on the radius of that circle, right?
If this is correct, then kudos for your article!
Thank you for this great explanation. I think it was the first time I really believed that radians were more practical than degrees. I radian measure is more useful when dealing with calculus, but now have a better explanation for my algebra 2 and trig. students as to why they should to learn it (other than it will help you in your future math classes).
@Jenn: Glad you enjoyed it! I was the same way, I didn’t appreciate what radians were until understanding their “point of view”, and it turns out, I think it makes more sense than degrees :). Happy to hear it will help with your teaching!
@Melissa: Awesome, so glad it helped! Yep, radians vs. degrees bugged me for years after school, happy you’re able to avoid that headache :).
@Mentock: Perhaps, not sure though. Far easier to count distance from your perspective [even number of paces or steps taken] than to figure out how far the observer moved their head.
this article is so eye opening! My math teacher always recommeds you as a second source after she gives us new material and now i understand why! I’ve never understood what the difference between degrees and radians were untill now! I really that that you put the definitions in simpler terms, “Degrees measure angles by how far we tilted our heads. Radians measure angles by distance traveled” !I will surely be dropping by again if i ever need other clarifications!