Wikipedia:Reference desk/Archives/Mathematics/2007 October 11

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October 11

Trignometry

Hello I wanted to ask that there are many a times the questions in trignometry dont come out with questions why is it so —Preceding unsigned comment added by 122.168.40.46 (talk) 10:29, 11 October 2007 (UTC)

I'm afraid I don't understand your question, but trigonometry might be useful. Algebraist 10:34, 11 October 2007 (UTC)

Perhaps the questioners want to say that many of the identities given at Trigonometry and List of trigonometric identities are presented without any justification as to why they should hold.  --Lambiam 16:05, 11 October 2007 (UTC)

I think the identities should have their proofs, with them. (If they don't currently), it would be nice if the question was clairified a bit more. A math-wiki 06:37, 12 October 2007 (UTC)

If you mean that there aren't any "questions" in the trig questions you're getting, it's because the problem set probably wants you to find all the angles and all the lengths and failed to tell you that these are the instructions. A lot of times in trigonometry you "solve" a triangle (ie give all 6 angles/lengths) based on the angles and side lengths you're given. There's no need to add a "question". (Indeed, it just confuses things: [1] —Preceding unsigned comment added by 84.0.126.201 (talk) 21:36, 12 October 2007 (UTC)

Argand diagram

This is, I confess, a homework question, but I'm not looking for an answer, just a route of attack. I've got to draw on an argand diagram, the set of all complex numbers z, such that $\arg {\frac {z}{z-1}}=x$, where x is an angle in radians (I'm keeping this question general). However, I can't see anyway of approaching this problem. Any hints on where to begin? Laïka 21:00, 11 October 2007 (UTC)

Try letting $z=a+bi\;\!$ where $a,b\in \mathbb {R}$, finding the real and imaginary parts of ${\frac {z}{z-1}}$, and writing an equation describing the fact that its argument is x. -- Meni Rosenfeld (talk) 21:26, 11 October 2007 (UTC)

Thanks, but could you be a bit more specific - I get: $\cos {x}+i\sin {x}={\frac {longformula}{otherlongformula}}$. Is there any way to solve this without having to perform the division, which just gives a long string of as and bs with no obvious representation on the Argand? Laïka 22:13, 11 October 2007 (UTC)

Hint: If you divide the imaginary part of ${\frac {z}{z-1}}$ by its real part, you get $\tan x\;\!$ (I understand that in your actual question x is some given number, so $\tan x$ is also a number). This should give you enough information about the relation between a and b to deduce the shape of the requested set. -- Meni Rosenfeld (talk) 22:22, 11 October 2007 (UTC)

Ah, yes, I see now. I also forgot that you could add arguments across a multiplication (and hence subtract across division). Many thanks! Laïka 13:47, 12 October 2007 (UTC)

Exponential problem

I recently came across the following question:

"Solve for x: 6^x+1 = (2^x+2)(3^2x)"

Upon trying, I found this question to be quite easy and got x=-1/2 as an answer. However, when I plugged it back into the equation, it didn't work. What I did was simply divide both sides by (3^2x) and easily got a common base of 2, then simply went on from there.

Could someone tell me how to solve this?

Thanks =). Acceptable 21:12, 11 October 2007 (UTC)

I assume that you mean $6^{x}+1=(2^{x}+2)3^{2x}\;\!$. The solution method you have described does not seem correct to me, so perhaps you should present it in more detail. I will also admit that I can think of no way of solving this equation, that I doubt there is any elementary way of solving it, and that for its numerical solution, -0.250447..., I can find no symbolic representation. -- Meni Rosenfeld (talk) 21:35, 11 October 2007 (UTC)

Oh no, I meant "6 to the power of x+1 = 2 to the power of x+2 times 3 to the power of 2x" —Preceding unsigned comment added by Acceptable (talk • contribs) 22:08, 11 October 2007 (UTC)

What I did was:

• 6^x+1 = (2^x+2)(3^2x)
• (6^x+1)/(3^2x) = (2^x+2)
• 2^-x+1 = (2^x+2)
• Therefore: -x+1 = x+2
• -2x = 1
• x = -1/2

But somewhere there, I must have made a mistake. Acceptable 22:05, 11 October 2007 (UTC)

You should use parentheses to specify the order of operations; the correct way to write this equation in plain text is 6 ^ (x+1) = 2 ^ (x+2) 3 ^(2x).

Your mistake is claiming that ${\frac {6^{x+1}}{3^{2x}}}=2^{-x+1}$. -- Meni Rosenfeld (talk) 22:12, 11 October 2007 (UTC)

Oh, am I able to simplify (6^(x+1)/(3^2x) any further? If not, what is the proper method that I should approach this problem with? Acceptable 22:27, 11 October 2007 (UTC)

Perhaps the article Exponentiation can help. The most relevant rules here are $(ab)^{n}=a^{n}b^{n}\;\!$ and ${\frac {a^{m}}{a^{n}}}=a^{m-n}$. -- Meni Rosenfeld (talk) 22:42, 11 October 2007 (UTC)

There is, quite possibly, a much easier way of looking at it. Instead of just using brute force (mathematically speaking), if we assume your only interested in real solutions, then we know that $6=2\cdot 3$, thus we con conclude that $x+2=2x$ and thus $x-2=0$ and so $x=2$ there probably exist nonreal solutions to this equation however, but I'm going to infer that your not interested in them (Em I right??). A math-wiki 23:07, 11 October 2007 (UTC)

$x=2\;\!$ is not a solution to our equation, which is $6^{x+1}=2^{x+2}3^{2x}\;\!$. -- Meni Rosenfeld (talk) 23:39, 11 October 2007 (UTC)

How could logarithms be employed to solve this? Acceptable 00:03, 12 October 2007 (UTC)

EDIT: Oh I see, I can split the 6 into (2x3), then distribute the exponent, this will leave with with identical bases on both sides. Thanks for your help. Acceptable 00:12, 12 October 2007 (UTC)

EDIT: Meni, I noticed that while thinking about it afterwards, now that I'm back home, the one problem with my solution is that it IGNORES $6^{x+1}$, which must also agree which must equal the other side when x=2, but the OP seems to have figured out how to use my original thinking appropriately. I assume the OP is thinking...

$2^{x+1}3^{x+1}=2^{x+2}3^{2x}$

Now $x+1=x+2$ has no solution, so the original equation doesn't have any solutions. Thus, $6^{x+1}\not =2^{x+2}3^{2x}$.

I should note that if the x on the left side of the equation was actually a y, then my solution would've existed. A math-wiki 06:15, 12 October 2007 (UTC)

You can't conclude from $2^{x+1}3^{x+1}=2^{x+2}3^{2x}$ that $x+1=x+2$. It looks like you're trying to apply a unique factorization rule but the best you can do with that is rule out integer solutions. In reals, it doesn't work. You need to do some logarithm work:

$2^{x+1}3^{x+1}=2^{x+2}3^{2x}$

$\log(2^{x+1}3^{x+1})=\log(2^{x+2}3^{2x})$

$\log(2^{x+1})+\log(3^{x+1})=\log(2^{x+2})+\log(3^{2x})$

$(x+1)\log 2+(x+1)\log 3=(x+2)\log 2+2x\log 3$

Now it's a linear equation in x. The rest is easy. It does have a solution. I won't spoil the ending, but it's between 0 and 1. --tcsetattr (talk / contribs) 07:57, 12 October 2007 (UTC)

I would personally work with the powers a little bit before taking logarithms:

$6^{x+1}=2^{x+2}3^{2x}\;\!$

$2^{x+1}3^{x+1}=2^{x+2}3^{2x}\;\!$

${\frac {3^{x+1}}{3^{2x}}}={\frac {2^{x+2}}{2^{x+1}}}$

$3^{1-x}=2\;\!$

And the rest is left as an exercise. -- Meni Rosenfeld (talk) 12:45, 12 October 2007 (UTC)

Correct, my solution is (or lack there of) is because I was only looking for integers. A math-wiki 01:00, 13 October 2007 (UTC)

Electrons on a Circle

I asked a similar question this yesterday, though this one seems much more complicated to me. I will go through my calucaltions so far, so if there are any errors, you can help me rectify them.

If you have n electrons distributed evenly around a circle radius r, distance x from another electron, what is the force of repulsion experienced by the electron not on the circle from the circle of electrons. Electrostatic repulsion is a inverse square law, and the coefficent of electrostatic force can be considered as k.

• r = radius (OA, OC)
• x = CY
• y = OB
• a = AY
• α = AOY
• β = BAY

using cosine rule; $a={\sqrt {x^{2}+(x+r)^{2}-2r(r+x)\cos \alpha }}$ so the force, being an inverse square is; $F={\frac {k}{x^{2}+(x+r)^{2}-2r(r+x)\cos \alpha }}$ now, seeing as always in the circle the components in y-direction will cancel, we can consider only those in the x-direction in order to caluclate a resultant force we need to consider only the x-directions components, to find this consider the triangle ABY, the angle β and distance OB are neccesary. using sine rule. $y=-r\cos(\alpha )\,$
therefore using sine rule,
$\sin \beta ={\frac {r-r\cos(\alpha )+x}{\sqrt {x^{2}+(x+r)^{2}-2r(r+x)\cos \alpha }}}$
so the x direction component of F, or Fsinβ is;
${\mathit {F_{x}}}={\frac {k(r-r\cos(\alpha )+x)}{{\big [}x^{2}+(x+r)^{2}-2r(r+x)\cos \alpha {\big ]}^{\frac {3}{2}}}}$

So to consider all the electrons; then $\alpha ={\frac {2\pi {\mathit {k}}}{N}}$ where N number of electrons. and k varies from 1 to N. Therfore a sum of all the forces would be.

${\mathit {F_{total}}}={\frac {k}{N}}\sum _{k=1}^{N}{\bigg [}{\frac {r-r\cos({\frac {2\pi k}{N}})+x}{{\big [}x^{2}+(x+r)^{2}-2r(r+x)\cos({\frac {2\pi k}{N}}){\big ]}^{\frac {3}{2}}}}{\bigg ]}$

Now this is it, I really need some help simplifying this massive mess, and like yesterday I really need the summation series out of the equation if that's possible. Thank you. ΦΙΛ Κ 22:04, 11 October 2007 (UTC)

I looks like to me; one, you have mistakenly assumed that the outside electron is in line with two electrons that are on opposite sides, and two, the question should be able to be answered by finding the relative locations of the circle electrons to eachother and then connecting the circle electrons to the outside one. A math-wiki 06:15, 12 October 2007 (UTC)

I have only assumed that the outside electron is inline with one electron, when there are either numbers, this does happen to become 2 oppostie electrons, however, this calculation is only for the purposes of a basic model, and is a simplification I am willing to make. —Preceding unsigned comment added by 85.189.108.163 (talk) 10:12, 12 October 2007 (UTC)

I'm guessing it's too difficult - however if there are 'infinite' electrons ie continuous the integral is do-able when integrating with respect to the distance from the centre (not angle - use change of variable 2D case of Shell theorem) - you probably already knew that.. Good luck waiting for a knight in shining white mathematical armour.87.102.87.36 12:23, 12 October 2007 (UTC)

Due to the fact its an inverse square law, the inverse square of the average distance isnt the same as the average of inverse square distances. 172.207.110.185 19:19, 12 October 2007 (UTC)

PS. A function that is a rough approximation will do! —Preceding unsigned comment added by 172.207.110.185 (talk) 19:26, 12 October 2007 (UTC)