Wikipedia:Reference desk/Archives/Mathematics/2011 October 21

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October 21

Trigonometry

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Okay so I have a homework question that I've been working on for a while and can't seem to solve. I have to verify that tan^2x-sin^2x=tan^2xsin^2x is an identity. So I tried working on the left side but I keep hitting dead ends. A nudge in the right direction would be helpful. Thanks. — Preceding unsigned comment added by Stj6 (talk • contribs) 00:10, 21 October 2011 (UTC)

Rewrite $\sin ^{2}x$ as $(\cos ^{2}x)(\tan ^{2}x)$. Then factor and apply a pythagorean identity.--130.195.2.100 (talk) 00:50, 21 October 2011 (UTC)

I understand how sin^2 is cos^2 times tan^2 but which sin^2 should I change. Also how would you factor (cos^2x)(tan^2x)? Thanks. Stj6 (talk) 01:12, 21 October 2011 (UTC)

Nevermind, I figured it out. Stj6 (talk) 01:18, 21 October 2011 (UTC)

isometry T => T - T(0) linear?

Let $T:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n}$ be an isometry. Prove that $T-T(\mathbf {0} )$ is linear.

How do you start with this? The (perhaps naïve) approach is to start with $T(a\mathbf {x} +b\mathbf {y} )-T(\mathbf {0} )$ and somehow manipulate that into $a\left(T(\mathbf {x} )-T(\mathbf {0} )\right)+b\left(T(\mathbf {y} )-T(\mathbf {0} )\right)$. Obviously you're supposed to take advantage of the fact that T is isometric somehow, but I don't see how. Perhaps someone can start by telling me that.Widener (talk) 07:33, 21 October 2011 (UTC)

This is the Mazur–Ulam theorem. Algebraist 07:57, 21 October 2011 (UTC)

I had a look at that proof, and I don't see why $\|gz-a\|=\|gz-ga\|$ I think really I was looking for a simpler proof. Widener (talk) 09:31, 21 October 2011 (UTC)

Because $g\in W$ and W is the family of all bijective isometries keeping the points a and b fixed, so $ga=a$.

To show that $T-T(0)$ is linear it is enough to show that for every $x,\ y$ and $a,\ b,\ a+b=1$ you have $T(ax+by)=aT(x)+bT(y)$. I think you can show this by considering the distances between $T(x),\ T(y),\ T(ax+by)$, and the triangle inequality. -- Meni Rosenfeld (talk) 11:20, 21 October 2011 (UTC)

How is that enough? Surely that only shows that $T-T(0)$ is linear if $a+b=1$

Suppose we were able to show that $T(ax+by)=aT(x)+bT(y)$ for $a+b=1$. It would then follow easily that $T(ax+by)-T(0)=a(T(x)-T(0))+b(T(y)-T(0))$ but again that is only for $a+b=1$ How does it show that $T(ax+by)-T(0)=a(T(x)-T(0))+b(T(y)-T(0))$ for all $a,b\in \mathbb {R}$? Widener (talk) 02:01, 22 October 2011 (UTC)

This is easy to show. Hint: Denote $S(x)=T(x)-T(0)$ and show that $S(ax)=aS(x)$.

Once you're finished you should read up on affine combinations, spaces and transformations, though I don't think the Wikipedia coverage is very good. Basically a transformation T is affine if $T(x)=v+S(x)$ where S is a linear transformation, and this is equivalent to "$T-T(0)$ is linear" and to "T preserves affine combinations (that is, $T(ax+by)=aT(x)+bT(y)$ for $a+b=1$)". -- Meni Rosenfeld (talk) 17:05, 22 October 2011 (UTC)

Alright, let's suppose we have deduced that $T(ax+(1-a)y)=aT(x)+(1-a)T(y)$ for all $a\in \mathbb {R}$. It then follows that $T(ax)-T(0)=T(ax+(1-a)0)-T(0)=aT(x)+(1-a)T(0)-T(0)=aT(x)-aT(0)=a(T(x)-T(0))$ What about the other linearity property $T(x+y)-T(0)=T(x)+T(y)-2T(0)$? The coefficients on the x and the y don't add to 1. Widener (talk) 21:59, 22 October 2011 (UTC)

But once you have $S(ax)=aS(x)$, you can scale coefficients that don't add to 1 as you please. If $a+b$ don't sum to one (0 is treated separately), let $\alpha ={\frac {a}{a+b}},\ \beta ={\frac {b}{a+b}}$ which do sum to 1, and then $S(ax+by)=(a+b)S(\alpha x+\beta y)=(a+b)(\alpha S(x)+\beta S(y))=aS(x)+bS(y)$. And then $S(x+y)=S(x)+S(y)$, etc.

Seriously, math isn't a spectator sport. We're not doing you any favor spoon-feeding answers all the time, you need to think for yourself about things. -- Meni Rosenfeld (talk) 06:02, 23 October 2011 (UTC)

I'm sorry Meni. It was late at night and I was tired and felt like going to bed and didn't really feel like doing math, so I thought I would post my query on the refdesk and go to sleep and hopefully someone would have responded in the morning. Usually I spend some time trying to figure it out myself first and I often succeed, but not always. Anyway, I have been able to solve the original question now (show T-T(0) is linear) thanks to the hints you provided, I appreciate it. Widener (talk) 07:11, 23 October 2011 (UTC)

You're welcome. -- Meni Rosenfeld (talk) 07:36, 23 October 2011 (UTC)

Since you're only looking at $\mathbb {R} ^{n}$, you can follow the linked proof up until it's justified why it suffices to preserve midpoints, and then argue that midpoints are preserved because tangent spheres only intersect in one point. Put another way, you only get equality in the triangle inequality when the extra point is along the segment connecting the first two points. This is a special property of the usual norm on $\mathbb {R} ^{n}$.--121.74.125.249 (talk) 11:50, 21 October 2011 (UTC)