## Wednesday, April 1, 2009

### Wine and Math

Who knew math would help you drink better? Enologix is a Napa Valley consultant using data analysis to help vineyards make better wine. I caught their profile in Wired. Not surprising in an industry dominated by connoisseurship, the analytic approach to production decisions has been quite controversial. I guess the only thing to do is open a bottle and find out. (You know the geeks are going to win, right. The geeks always win.)

pic by Buson

Anonymous said...

Funny, I'm on my fourth glass of merlot, trying to figure out an easier way to do this problem besides finding an LCD and grinding it out that way:

Prove the identity:
(Pre-Calculus 5th ed., James Stewart.
chp. 7.2 #40)

tan(x-y)+tan(y-z)+tan(z-x)=tan(x-y)tan(y-z)tan(z-x)

I've used substitution to say:

tan x = a
tan y = b
tan z = c

just to make the handwriting legible. But that doesn't matter.....

but, does tan(-x) = -tan x mean anything? Can I apply that to.....anything in this problem?

-j

Unknown said...

After 4 glasses of Merlot, I would be drooling at this identity. I think I'd try 4 cups of coffee.

Tinyc Tim said...

Prove that

tan(x-y)+tan(y-z)+tan(z-x)=tan(x-y)tan(y-z)tan(z-x)

Let a=x-y b=y-z

Since -(a+b)=z-x

tan(a)+tan(b)+tan(-(a+b))=tan(a) tan(b) tan(-(a+b))

Divide thru by tan(a+b) and use fact that tan(-(a+b))=-tan(a+b)

(tan(a)+tan(b))/tan(a+b)-1=-tan(a)tan(b)

(tan(a)+tan(b))/tan(a+b)=-tan(a)tan(b)+1

Take reciprocals

(tan(a+b))/(tan(a)+tan(b))=1/(1-tan(a)tan(b))

Multiply thru by tan(a)+tan(b)

tan(a+b)=(tan(a)+tan(b))/(1-tan(a)tan(b))

The above is a well-known identity.

Anonymous said...

Just so you know, they were small wine glasses, and I only fill them halfway to keep them from spilling (and to drink less!)

I still need to go back to Lee's example and see if I can harness it.

-j

Tinyc Tim said...

OK Jeff - The latest one you are working on, which we may discuss tonight during our tutoring session, is discussed at Yet Another Trigonometric Identity Proof. Sorry, but you're going to have to decipher my handwriting this time.

- Lee

Anonymous said...

Lee,

I'm thrilled there's a web address with my own personal identity!

Clever work with the area proof :)

-jeff

Tinyc Tim said...

This is simply a copy / paste of an email exchange between Jeff and me on April 21/22. It fits in w/ trig identities (sort of) so I thought it belonged here.

Here's the exchange:

My suspicion that you are smart and persistent has been confirmed. Your proof is flawless. I also like your notation; math is a notational pain so I liked things like sinx in place of sin(x) - easier to write and clear what is meant.

Thanks so much Jeff for sending this and for being so forgiving when a teacher who is supposed to help sometimes is unable to deliver. The student who preceded you was throwing trig identities at me that I stumbled on as well. He too was understanding. I did manage to say a few things that shed some light on his problems but ... I fear he left w/o getting his money's worth. I know - you don't pay but you know what I mean.

Oh yeah. When you find out more about that Stat's proportion / confidence interval mystery you couldn't resolve, teach me that too.

BTW, you're conversing w/ someone who is 67 1/365 years young today. I am finally, again, "in my prime." It's been a long (6 year) stint being composite and it's good to be back.

I'm going to send your stuff to yofx if you don't mind.

Well done, sir.

Lee

(Jeff's problem and its solution follow):

I knew I was on the right track with factoring it out somehow, but alas, we ran outta time. Here was my next attempt:

2sinxtanx - tanx = 1 - 2sinx

{move everything over to the left side. hmm, this looks familiar}

2sinxtanx - tanx + 2sinx - 1 = 0

{grouping}

(2sinxtanx - tanx) + (2sinx - 1) = 0

{factoring}

tanx(2sinx-1) + (2sinx-1) = 0

{factoring again}

(tanx + 1)(2sinx - 1) = 0

{solving!}

tanx + 1 = 0 and 2sinx - 1 = 0

tanx = -1 and sinx = 1/2

Notice how that slick move you liked did end up being involved in the answer? But unfortunately, multiplying across the equal sign by an unknown quantity may lead to things dropping out, as it did when my illegal way "lost" the sinx = 1/2

So now, the tangent is a -45 degrees (or -pi over 4). In the fourth quadrant, this makes the angle 7pi over 4 and in the second quadrant, it is 3pi over 4

The sine is a special angle, 30 degrees (or pi over 6), so in the first quadrant, the answer is pi over 6, and in the second quadrant, the answer is 5pi over 6.

Look good? I think everything worked right.

-jeff