Wikipedia:Reference desk/Archives/Mathematics/2011 July 19

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July 19

trigonometric identities

Hi, can anyone help me with the problem of proving the identity of: (2sec^2x+tanx-8)/sec^2x-5 == (2sinx-3cosx)/sinx-2cosx

I had already tried a couple of things, like converting the left side into tan: [(2tanx-3)(tanx+2)]/tan^2x-4

Or converting it into sec: (2sec^2x+sec^2x-7)/sec^2x-5

But either way I'm stuck. Can anyone help me? --Thebackofmymind (talk) 10:38, 19 July 2011 (UTC)

Just to rewrite it, you want to show that

${\frac {2\sec ^{2}x+\tan x-8}{\sec ^{2}x-5}}={\frac {2\sin x-3\cos x}{\sin x-2\cos x}}\,.$

Have you tried replacing tan(x) with sin(x)/cos(x) and sec(x) with 1/cos(x)? — Fly by Night (talk) 11:24, 19 July 2011 (UTC)

I think you're on the right track using tan, you've just missed a step. Whilst you've factorised the numerator as (2tanx-3)(tanx+2), the denominator (tan^2x-4) can also be factorised using the difference of two squares principle. That should put you on the right track; as FbN suggests, there'll be a point to use the identity of tan as sin/cos. Grandiose (me, talk, contribs) 11:30, 19 July 2011 (UTC)

Oh, I see it now. Thank you both, Fly by Night and Grandiose! (By the way, do you know why there's always a period at the end of these equations?)--Thebackofmymind (talk) 11:47, 19 July 2011 (UTC)

Equations appearing in text should be punctuated according to the standard rules of English punctuation. So if an equation appears at the end of a sentence, as is often the case, the equation itself should be followed by the apprpriate punctuation mark (perhoiod, question mark, etc.) Sławomir Biały (talk) 12:35, 19 July 2011 (UTC)

Thank you Slawomir Bialy. --Thebackofmymind (talk) 14:21, 19 July 2011 (UTC)

Help explain notation in Mathematical Logic book

Could somebody with a background in mathematical logic please explain what is being said by the following notation in Exercise 4.6 on p.23 of Ebbinghaus' Mathematical Logic? He introduces the notation on p. 19 but not in a way that obviously generalizes to having two things in the bottom of the "fraction". How should I interpret

${\cfrac {}{x\quad x}}\qquad {\text{and}}\qquad {\cfrac {y\quad t_{i}}{y\quad ft_{1}\ldots t_{n}}}$

I've tried various ways and none of them seem to make perfect sense. The rules somehow correspond to Definition 4.5 (a) given directly above. This is not a "homework question". I'm just a self-learner. I know Wikipedia isn't a discussion board too but I don't know where else this question might get answered. Jason Quinn (talk) 13:41, 19 July 2011 (UTC)

That is not a fraction. It is a premise and conclusion (or an if-then) statement. In the first, there is no premise. It is simply stating that "x x" has some property (call it P). The second states that if "y t_i" has the property P, then "y f t_i ... t_n" also has the property P. -- kainaw™ 15:53, 19 July 2011 (UTC)

Thanks. My confusion was over what the "$x\quad x$", "$y\quad t_{i}$", and "$y\quad ft_{1}\ldots t_{n}$" were. It appears that they are simply strings with the large spacing still indicating concatenation so the actual spacing is immaterial and only used for ease of reading. I was interpreting it to be some sort of qualification on "x" as these variable-variable and variable-term combinations are neither terms nor formulas. That threw me off for a bit. I also believe the "y" to be a variable even though it was never explicitly defined as "x" was. Under my present understanding, the calculus is "really" the S-term calculus plus these rules:

${\cfrac {}{xx}}\qquad {\text{and}}\qquad {\cfrac {yt_{i}}{yft_{1}\ldots t_{n}}}$

Thanks again for your reply. Jason Quinn (talk) 01:26, 20 July 2011 (UTC)

Date calculation problem

Due to issues beyond my control, I don't have time to think this through, but I thought this might be a bit of a puzzle that the math desk regulars would like - and it is a real problem, not one of those made up "I have a puzzle" problems... You are given two dates. You have year, month, and day of week (ie: 1=Sunday, 2=Monday, 3=Tuesday...). You do not have the day of the month. You are also given the number of days between the two dates (which includes both dates - so if they are the exact same date, you get 1 for length of time). Is it ever possible to calculate the exact dates - how? How about getting just 2 choices for each date? Will there be times when there are 5 choices for both dates? I will be working on this in a few hours and will post what I find, but I'm sure someone here will have a better algorithm than what I come up with. -- kainaw™ 14:26, 19 July 2011 (UTC)

I have discovered that in the actual data, the length between the two dates tends to be 5 to 10 days. As such, it rarely falls between two different months. So, within a single month, 5 days between, say, a Monday and Friday will occur at least 4 times. Calculating actual dates is a comparatively rare occurrence. -- kainaw™ 15:47, 19 July 2011 (UTC)

Why is the length of time between date (A) and date (B) 1 when A=B? hydnjo (talk) 23:30, 19 July 2011 (UTC)

Length of time (number of days) is inclusive. It includes A and B. So, if A=B and length includes A and B, the length is 1. -- kainaw™ 02:34, 20 July 2011 (UTC)

Sure, there are plenty of times when there are 5 solutions. For example, if A is July 2011, Friday and B is July 2011, Friday, and the number of days between (inclusive) is 1, then the solutions are the 1st, 8th, 15th, 22nd and 29th. Raise the number of days to 8, and you drop to 4 solutions. Raise to 15, 22 or 29 for 3, 2, or 1 solutions, respectively.--Antendren (talk) 03:58, 20 July 2011 (UTC)

29 will more typically have 4 solutions (when the two dates are in different months). Similarly for 22 with 3 solutions. Cases where you have just 1 solution are fairly rare even if the gap is large. Rckrone (talk) 18:14, 20 July 2011 (UTC)

I originally thought this would be an interesting problem, but when I found that nearly all of the data had a length between 1 and 5, I realized that it wouldn't be possible to match it up to real dates with any confidence. On the relatively rare instance that it spans two different months, I can line it up, but there's another catch that I discovered - it is rounded to the nearest number of days. So, if the time between A and B is less than 12 hours, the number of days is 0. If it is between 12 and 26 hours, the number of days is 1. That rounding down thing without having access to the time for A and B makes it just a little bit harder. My end algorithm is this (in case anyone is interested):

• Given: Ayear, Amonth, and Adow (Day of Week). I want to calculate Adom (Day of Month)
• Given: Dyear, Dmonth, and Ddow. I want to calculate Ddow. (I'm using D because the real data uses A and D).

1. Check to ensure Ayear<=Dyear. If not, data is bad.
2. Check to ensure that if Ayear=Dyear, then Amonth<=Dmonth. If not, data is bad.
3. Set Adom=1.
4. While the dow for Ayear-Amonth-Adom is not the same as Adow, increment Adom by 1 (keep adding a day until A is on the correct day of the week)
5. Set Ddom=1.
6. While the dow for Dyear-Dmonth-Ddom is not the same as Ddow, increment Ddom by 1.
7. While Dyear-Dmonth-Ddom comes before Ayear-Amonth-Adom, add 7 to Ddom (Make sure D comes after A by adding 1 week to D until it comes after A).
8. While (D-A)>=length+1, add 7 to A. In this, D-A is the number of days between D and A. If it is too long, move A up by a week. I add 1 to length because of that rounding thing the real data has.
9. While (D-A)<=length-1, add 7 to D. In this, I am moving D a week into the future when D-A is too short. I subtract 1 from length to get around the rounding error here.

I have been cross-referencing with other data and this isn't doing too bad. In general, time between A in one event and A in another event is off by an average of 1.5 weeks from what the perfectly accurate data shows. -- kainaw™ 20:06, 20 July 2011 (UTC)