Wikipedia:Reference desk/Archives/Mathematics/2011 March 6

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March 6

Trig identities

Hi. If α=sin(2θ), then how would I express sin⁴(θ) + cos⁴(θ) in terms of α? I have tried various combinations of the identities $\sin ^{2}\theta ={\frac {1-\cos \theta }{2}},\cos ^{2}\theta ={\frac {1+\cos \theta }{2}},\sin(2\theta )=2\sin \theta \cos \theta ,\cos(2\theta )=\cos ^{2}\theta -\sin ^{2}\theta =1-2\sin ^{2}\theta =2\cos ^{2}\theta -1$, all without getting myself anywhere useful. Any help would be appreciated. (PS: This is not homework, this was in a puzzle that I did a few days ago) 72.128.95.0 (talk) 01:44, 6 March 2011 (UTC)

The trick to dealing with trigonometric identities without resorting to Euler's formula is to massage everything into a form where you can use $\sin ^{2}\theta +\cos ^{2}\theta =1$ to eliminate one of the trig functions in favor of the other. So take $\alpha =2\sin \theta \cos \theta$ and square it; then express the right-hand side in terms of $u=\sin ^{2}(\theta )$. Your goal $\sin ^{4}\theta +\cos ^{4}\theta$ can also be expressed in terms of $u$. Do these two expressions look related? –Henning Makholm (talk) 02:10, 6 March 2011 (UTC)

Take a lot at $\sin ^{2}(2\theta )$ and $\cos ^{2}(2\theta )$ (using the identities you mention) and see if you can combine them to get what you want. --Tango (talk) 02:15, 6 March 2011 (UTC)

We know that sin²θ + cos²θ = 1 for all θ, and so squaring both sides tell us that (sin²θ + cos²θ)² = 1². Expanding gives

$\sin ^{4}\!\theta +2\sin ^{2}\!\theta \cos ^{2}\!\theta +\cos ^{4}\!\theta =1\,.$

In your notation α = sin(2θ). However, sin(2θ) = 2.sin(θ).cos(θ), and so α² = 4.sin²θ.cos²θ. It follows that

$\sin ^{4}\!\theta +\cos ^{4}\!\theta =1-\left(\alpha ^{2}/2\right).$

Don't feel put off. Almost all students have problems with trigonometric equalities like this at one time or another. There's a knack to it. It often feels more like an art that a science. — Fly by Night (talk) 03:54, 6 March 2011 (UTC)

As an aside, this seems like a nice problem for its applications. For example, –1 ≤ sin(2θ) ≤ 1 and so 1/2 ≤ 1 – α²/2 ≤ 1; meaning that 1/2 ≤ sin⁴θ + cos⁴θ ≤ 1 for all θ. — Fly by Night (talk) 04:39, 6 March 2011 (UTC)

A simple-looking but not easy equation

Are there any nontrivial integer solutions to

$a^{b}-b^{a}=a+b$

other than a=2, b=5?

Newyorkbrad (talk) 03:18, 6 March 2011 (UTC)

Plotting it strongly suggests no (though transforming what the plot shows to a formal proof is surprisingly hard). –Henning Makholm (talk) 05:48, 6 March 2011 (UTC)

(edit conflict) I doubt that it does. I ran a quick check on my laptop for 1 ≤ a < b ≤ 10,000 and the only solutions that turned up were (a,b) = (2,5) as you already mentioned. Plotting the set x^y – y^x – x – y = 0 for real x and y (not just integers) with 1 ≤ x ≤ y seems to suggest that the curve has an asymptote with an irrational slope. Given that a straight line with an irrational slope can have at most one point with integer coordinates, it would suggest that as a and b get larger, there is less and less chance of finding solutions. Although, this is totally un-rigorous, and is just a gut feeling. — Fly by Night (talk) 05:52, 6 March 2011 (UTC)

I'm quite sure the asymptote is the line x=y, which is an even better argument, because the curve never reaches it precisely, yet gets close enough to have no chance of hitting an integral point off the asymptote. Anyway, here's something more rigorous:

We can define "non-trivial" to mean a and b both positive. It is easy to see that the only solution where one of them is 1 is $1^{0}-0^{1}=0+1$ which is trivial, so wlog either of them is at least 2.

First we prove that non-trivial solutions requires a<b. It is necessary at least that $a^{b}-b^{a}$ is positive. Set q=a/b, assuming for a contradiction that q>1, and calculate:

$(qb)^{b}>b^{qb}$  ⇒  $b(\log q+\log b)>qb\log b$  ⇒  $\log q>(q-1)\log b$  ⇒  ${\tfrac {\log q}{q-1}}>\log b$

But the left-hand side of the last inequality is at most 1 for q>1, so $\log b<1$, so b must be 2. And a must be 3 (or ${\tfrac {\log q}{q-1}}$ would drop below $\log 2$), but a=3, b=2 is not actually a solution. Therefore a<b, and we write b=a+n for some positive n.

We know one solution with a=2 and it is obviously the only such one. So assume that a≥3. Then

$a^{a+n}-(a+n)^{a}=a^{a}{\big (}a^{n}-{\big (}{\tfrac {a+n}{a}}{\big )}^{a}{\big )}=a^{a}{\big (}a^{n}-e^{a\log(1+{\frac {n}{a}})}{\big )}>a^{a}(a^{n}-e^{a{\tfrac {n}{a}}})=a^{a}(a^{n}-e^{n})$

which is clearly increasing with n, as $a>e$. For a=3 we can see by direct computation that there is no solution before $a^{a}(a^{n}-e^{n})$ gets too large, and for a≥4 we always have $a-e>1$, so even n=1 is too much because $a^{a}$ is clearly larger than a+(a+1). So there is no solution with a≥4 either. –Henning Makholm (talk) 07:18, 6 March 2011 (UTC)

Good job! — Fly by Night (talk) 15:03, 6 March 2011 (UTC)

Many thanks to both of you! I don't know if this problem has ever been published anywhere; I haven't seen it (but then I'm not a mathematician or math educator so the odds are I wouldn't). Regards, Newyorkbrad (talk) 03:18, 7 March 2011 (UTC)

What's wrong with a=0, b=0? Or is that deemed trivial? -- SGBailey (talk) 08:44, 7 March 2011 (UTC)

That would be trivial, and the OP asked for "nontrivial integer solutions". — Fly by Night (talk) 18:53, 7 March 2011 (UTC)

Also, that depends on the value of 0⁰; where I come from, it's undefined. PhageRules1 (talk) 20:06, 11 March 2011 (UTC)

Quick query - showing a simplicial complex has trivial 2nd Homology group

Hi all,

If I've triangulated the real projective plane RP^2 as follows:

www.math.jhu.edu/~jmb/note/rp2tri.pdf

and am treating that as a simplicial complex, K, then how can I argue (assuming I'm correct and it does) that H_2(K) is the trivial group? I tried to make some argument about orienting the ten 2-simplices in the complex (label as shown, then suppose ABD is positively oriented in a chain which is a 2-cycle...) but I couldn't get anything out. Am I being stupid? I'm fairly new to all this orientation stuff, but it seems like i should be able to make a fairly straightforward argument where, e.g., we can't orient the middle triangle EDF because the surrounding triangles have different orientations. However, maybe it's too early in the morning and I'm jut being slow.

My friend says if ABD is +vely oriented, then so are BED, ADC and BAF, thus so is CDF and so is FAE, which is our contradiction. However, I don't see why if ABD is positively oriented, that means BAF is too. Could anyone please help?

Thank you!

Typeships17 (talk) 10:38, 6 March 2011 (UTC)

If any triangle in the cycle is positively oriented, then all of its neighbors must be as well in order for the cycle to be closed. Thus you can conclude that all of the triangles in the picture are positively oriented. But this is impossible because edges that are diametrically opposed on the boundary must carry the opposite orientation. Sławomir Biały (talk) 12:54, 6 March 2011 (UTC)

Got it, thank you very much! :) Typeships17 (talk) 14:40, 7 March 2011 (UTC)

Direct computation is very easy for RP². Draw a circle, and put a vertex at 3 o'clock and 9 o'clock, labelling both vertices as a. Then label both the upper and lower semi-circular arcs as A and give them both clockwise orientations. Finally, label the face α with a clockwise orientation. This gives the standard model for the projective plane (although it is a CW complex instead of a simplicial complex). Next consider the sequence of boundary maps:

$0{\stackrel {\partial _{0}}{\longrightarrow }}F{\stackrel {\partial _{F}}{\longrightarrow }}E{\stackrel {\partial _{E}}{\longrightarrow }}V{\stackrel {\partial _{V}}{\longrightarrow }}0\,,$

where F corresponds to the faces, E to the edges and V to the vertices. Clearly Im(∂₀) = 0 and Ker(∂₀) = 0. Next, ∂_F(α) = 2A, so Im(∂_F) ≅ 2Z while Ker(∂_F) = 0. Then ∂_E(A) = a − a = 0, meaning that Im(∂_E) = 0 and Ker(∂_E) ≅ Z. Finally, since ∂_V(a) = 0 it follows that Im(∂_V) = 0 and Ker(∂_V) ≅ Z. The homology groups are given by H₂ ≅ Ker(∂_F) / Im(∂₀), H₁ ≅ Ker(∂_E) / Im(∂_F), and H₀ ≅ Ker(∂_V) / Im(∂_E). Putting that together gives:

$H_{2}(\mathbf {RP} ^{2},\mathbf {Z} )\cong 0,\ H_{1}(\mathbf {RP} ^{2},\mathbf {Z} )\cong \mathbf {Z} _{2},\ H_{0}(\mathbf {RP} ^{2},\mathbf {Z} )\cong \mathbf {Z} \,.$

If you only wanted H₂(RP¹,Z) then you only need look at ∂₀ and ∂_F. — Fly by Night (talk) 19:16, 7 March 2011 (UTC)