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$\frac 12 x \sin x=1$ . Let's look at a right triangle with base $x$ and altitude $\sin x$ . Then our equation is for the area of this triangle. Let the sides of the triangle be $a=x$ , $b=\sqrt {x^2+sin^2 x}$ , and $c= \sin x$ . According to wikipedia, Heron's formula can be written as $$A=\large \frac { \sqrt {4a^2c^2-(a^2+b^2-c^2)^2}}{4}$$

Plugging in:

$4=\large \sqrt{4x^2 \sin^2 x-(x^2+x^2+\sin^2 x-\sin ^2 x)^2}$

$4=x^2 \sin^2 x -x^4$

$x^2(x^2- \sin^2 x)=-4$

$x^2$ will always be positive, and $\sin^2 x$ is never greater than $x^2$ , so this equation can have no real solutions. The original has solutions, so why is this wrong?

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2  
an application of the intermediate value theorem to say $x=0$ and $x=12\pi +\pi/2$ shows there must be a solution. –  James Jul 12 '13 at 23:33
    
Yea I know wolframalpha says there are infinitely many –  Ovi Jul 12 '13 at 23:34
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draw a graph of $y = x \sin x,$ say for $0 \leq x \leq 4 \pi.$ –  Will Jagy Jul 12 '13 at 23:36
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I think this particular wikipedia formula is wrong, probably it is $a^2-b^2+c^2$ correctly if starts with $4a^2c^2$. Try it out with arbitrary right triangle. There's negative under the square root in your case!! –  Berci Jul 12 '13 at 23:47
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I've updated the formula on Wikipedia, after verifying in Mathematica that Berci's comment is correct. –  Rahul Jul 12 '13 at 23:55

2 Answers 2

up vote 6 down vote accepted

This particular wikipedia formula is wrong.

It should be correctly either $$A=\large \frac { \sqrt {4a^2b^2-(a^2+b^2-c^2)^2}}{4}$$ or $$A=\large \frac { \sqrt {4a^2c^2-(a^2-b^2+c^2)^2}}{4}\,.$$ Mind the symmetry..

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3  
The formula on Wikipedia is correct, but the OP copied it down incorrectly. –  wj32 Jul 12 '13 at 23:56
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@wj32 I think I copied it down correctly but Rahul Narain said he just updated it on wikipedia. –  Ovi Jul 12 '13 at 23:58
    
Well, my browser (still) shows the cited version of the formula. However, Rahul Narain commented that he already corrected it in wiki.. –  Berci Jul 12 '13 at 23:58
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Yeah, Wikipedia was updated today, no doubt as a result of this problem. –  Thomas Andrews Jul 12 '13 at 23:58
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@Ovi: With $a=x$, $b=\sin x$, $c=\sqrt{x^2+\sin^2x}$, the formula just gives you back $\frac12x\sin x$. Not surprising, since we already know that's the area of the triangle... –  Rahul Jul 13 '13 at 0:00

There are solutions on each interval $\left[2k\pi,2k\pi+\frac\pi2\right]$ for positive integer $k$ by the intermediate value theorem because $x\sin(x)$ is $0$ on the left end and $2k\pi+\frac\pi2$ on the right.

Heron's Formula should be $$ A=\frac{\sqrt{4a^2b^2-(a^2+b^2-c^2)^2}}{4} $$ Does that cause the same problem?

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Probably, but I will have time later to check –  Ovi Jul 13 '13 at 0:01

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