Wikipedia:Reference desk/Archives/Mathematics/2012 March 27

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

March 27

Trigonometry practice questions

Is there some websites where I can practice trigonometry like right-angled triangle and Pythagorean theorem, sine, cosine, tangent, cosecant, secant, cotangent, degrees, minutes and second form, angle of elevation, angle of depression, trigonometric functions of any angle, converting between degree and radian measure, graphs of the sine, cosine, and tangent, applications of trigonometry; law of sines and law and cosines? — Preceding unsigned comment added by 70.54.64.180 (talk) 01:09, 27 March 2012 (UTC)

All I can come up with is a search for "Trigonometry exercises", I'm guessing you can do that as well. Some results anyway:

http://math.usask.ca/emr/menu_trig.html

http://www.analyzemath.com/trigonometry_worksheets.html

http://www.homeschoolmath.net/online/trigonometry.php

math.bard.edu/belk/math141/TrigonometryExercises.pdf

http://www.khanacademy.org/exercise/inverse_trig_functions Ssscienccce (84.197.178.75) (talk) 17:51, 27 March 2012 (UTC)

Help me name or find the existing name for this geometric concept!

This may have a proper name, if so - let's discuss. If not, let's name it. This is for a web application in C#, so whatever we call it I will start naming as such in my code.

I'm taking GPS data as a collection of n 'Points', each point has latitude (degrees), longitude (degrees), elevation (meters), and datetime(DateTime object). From this, I compute a list of n-1 'Segments.' In each Segment object, I have distance(meters), time(seconds), and velocity(meters/second), effectively taking the derivative between each consecutive point.

Now, I want to n-2 '<name this object>s'. Each will have acceleration (meters/second/second). A good name for a collection of segments may be 'Path' or 'Track', but this is specifically an object that is two segments only (comprised from three consecutive points), which holds the second derivative value.

I'm leaning toward 'Segue' and this will be the object's name as I flesh out the details. Is there an existing name for this concept, or does anyone have any suggestions? — Preceding unsigned comment added by Ehryk (talk • contribs) 03:42, 27 March 2012 (UTC)

I'm not quite following. Why are your computing such values ? As for acceleration, wouldn't you also need an initial velocity (speed and direction) in order to ensure that you go through all 3 locations at the indicated times ? Or do you assume the initial velocity is always zero ? (If so, some combos of 3 locations and times might not work.) And don't you also want to know the velocity at points 2 and 3, as well ? Also note that your acceleration will be a 3D vector, so will have i, j, k components.

Perhaps I'm not quite following. I am pulling the points from GPS data which stores all the points, I'm analyzing it to determine things like max acceleration, average velocity, etc. I don't assume initial velocity is zero, nor am I trying to find the velocity AT any point, or the acceleration AT any segment. I'm computing velocity BETWEEN points, and acceleration BETWEEN segments, and looking for the best name for 'Between two connected segments.' Ehryk (talk) 15:04, 27 March 2012 (UTC)

In the first case, you determine the average velocity, which has some value. However, in the second case you're calculating the average acceleration, which doesn't seem of much use to me, say, when calculating a car trip, since the acceleration is so uneven (sometimes positive, negative when you brake, sometimes zero). Since periods of accel and decel are so short, the three GPS coords would have to be taken within maybe a second of each to give a good approximation for current accel, and at such resolution, inaccuracies in the GPS points would likely throw it off. So, is your application designed to find car accel, or something else ? StuRat (talk) 17:02, 27 March 2012 (UTC)

Why does uneven lead you to unuseful? The time resolution can be set by the individual GPS units, but on a Garmin under 'Fine' update mode we're getting points every 0.5 - 3 seconds. I can then compile these values into 'Time spent braking', 'Time spent coasting', and 'Time spent accelerating', 'Max acceleration', 'Max deceleration', etc. - each of which seem useful to me.Ehryk (talk) 19:21, 27 March 2012 (UTC)

Well, even at the 0.5 second increment, that's a second elapsed for 3 points, and the accel may change quite a bit over a second. Also, as I said before, the error in the GPS coords must be significant in such tightly spaced points. Do you know what the accuracy is ? I'd think, to get accurate results for instantaneous accel, you'd want to use an accelerometer. StuRat (talk) 04:24, 28 March 2012 (UTC)

You're quite right, but the point of this isn't really for accuracy with regard to instantaneous acceleration, average will do just fine. All I'm trying to glean is more basic analysis: when were you, on average, increasing in speed, decreasing, staying roughly the same, and in the context of driving in this application, if you can't hold your accel/decel for over a second, I don't want it.Ehryk (talk) 09:19, 28 March 2012 (UTC)

Without knowing anything else, I'd want to call the first group 'Speed' and the second group 'Accel' (even though more info is stored than just that). Or you could use the 3 point analogue to a 'Segment', and call the second group an 'Arc'. StuRat (talk) 05:44, 27 March 2012 (UTC)

"Arc" sounds like a good word for this, though I don't know of a standard term for what you're describing. Or you may want to use Seg1, Seg2 and Seg3 for points, segments and arcs. -- Meni Rosenfeld (talk) 12:38, 27 March 2012 (UTC)

I'm looking for the name of the geometric concept that will contain speed, acceleration, distance values, etc. Segment is precisely what I want for the first derivative. 'Arc' doesn't sit right with me, because I immediately think of an arc of a circle, so I'd store things like 'radius, start, end, degrees' in that. I can go with new terms here, as well - if there isn't a term that precisely describes this already. Ehryk (talk) 15:04, 27 March 2012 (UTC)

Note that an arc isn't necessarily circular. For example, there are also arcs described by polynomial equations. See arc (geometry). StuRat (talk) 17:05, 27 March 2012 (UTC)

If you want to focus on the betweenness, you can use the adjectives interstitial [1] or interpolated, so maybe "interstitial velocity" would be a good word for an estimated velocity between two known velocities. If you want to focus on the "completeness" of a set of values for acceleration, velocity, etc, you could try something with telos, which is really just greek for the end. As in, once you know all these variables in a deterministic Newtonian system, you know the telos of the object. I also like Meni's suggestion of arc. It (and telos) is often used in a sense of "trajectory" or "ultimate path", and is not limited to circles. SemanticMantis (talk) 16:51, 27 March 2012 (UTC)

Arc was StuRat's suggestion, I merely concurred. -- Meni Rosenfeld (talk) 18:33, 27 March 2012 (UTC)

Thanks, I appreciate these and I'll consider them. I was hoping for a more definite term, like how Segment immediately evokes connecting two points, that would mean two connected segments in itself. It's good to note that there is not a term that means this.Ehryk (talk) 19:21, 27 March 2012 (UTC)

not so fast, Ehryk. There's the concept span, which in bridging and other fields can be thought to be composed of segments. Travel span analysis? --Tagishsimon (talk) 19:27, 27 March 2012 (UTC)

Right, but if I wanted 'composed of segments' I could use: Span, Path, Track, Polygon (if the ends connect), Journey... and others. I don't mean "Contains one or more segments", I want a word: A geometric region is an <xxxx> iff it is TWO segments connected such that they share an endpoint. Unless we're all mistaken, there isn't a word that means specifically this.Ehryk (talk) 19:40, 27 March 2012 (UTC)

Note that the term spline is used to describe two or more arcs blended together (and those arcs can include line segments, as in the first illustration in that article). StuRat (talk) 04:18, 28 March 2012 (UTC)

I'd have to say that's the closest I've come to an answer. Keep in mind, I'm up for made up words, too: Bisegment, DualSegment, SegAngle, Kink, Turn, etc. etc. - throw some new ideas out!Ehryk (talk) 09:19, 28 March 2012 (UTC)

Vertex ? Corner ? StuRat (talk) 03:23, 31 March 2012 (UTC)

Homomorphisms from GL(n,Z) into a symmetric group?

Fix n > 1. Other than the homomorphism induced by looking at determinants, are there any non-trivial homomorphisms from GL(n,Z) into a (non-trivial) symmetric group? The determinant map is the only one I've been able to come up with. I know generators for the automorphism group of GL(n,Z), and have tried pre-composing the determinant map by these, but they either don't change it or make the map trivial. It's such a nicely phrased question, surely someone must have tackled it before...? Thanks for any tips! Icthyos (talk) 14:20, 27 March 2012 (UTC)

Actually, it turns out the only target group I'm interested in is the group of order two. I'm thinking it might work to show that any homomorphism from GL(n,Z) to Z₂ has the property of being a determinant-like function, then invoking uniqueness of such a function to show the only homomorphism is Det. Does that sound plausible? Icthyos (talk) 21:03, 27 March 2012 (UTC)

The problem is equivalent to finding all normal subgroups of finite index. The mod n subgroups provide a great many examples. The normal subgroups have been classified to some extent by Newman in 1967. Sławomir Biały (talk) 22:16, 27 March 2012 (UTC)

Ah-hah, what a clever way of thinking about it. I forgot about Cayley's theorem. Thanks! Icthyos (talk) 18:25, 28 March 2012 (UTC)

Congruence subgroup may be of interest for Sławomir's suggested examples, though that article needs more on what the solution of the problem was, mostly, those are the only examples. Especially for the case of maps into Z₂ or an abelian group, see also Algebraic_K-theory#K1 for the infinite general linear group over Z. Milnor's book IIRC should have enough on the finite-dimensional (sub)groups you are particularly interested in. (always approximate the finite by the easier infinite :-) ). The point is that these groups are simple enough, noncommutative enough, that for maps to abelian groups, only the determinant should be there. I.e. the commutator subgroup should be generated by the elementary matrices and should be all of SL. (see also Special_linear_group#Relations_to_other_subgroups_of_GL.28n.2CA.29 and other linked articles here and there).John Z (talk) 10:17, 1 April 2012 (UTC)