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[Axxle]

It's always nice to finally understand why something I knew was true.  For instance, I learned a while ago that e^(i*pi) is equal to -1, but I never knew why.  In a calculus textbook I was reading, he went off on a random tangent (not a sine (but now I might be going off on a cotangent mentioning this)) about imaginary exponents, and I realized what was going on while it was happening.

I think that's a big reason people don't like math early on, everything just seems so arbitrary. It's when you get into the reasons behind why formulas work the way they work when it clicks.

[Kirian]

It's always nice to finally understand why something I knew was true.  For instance, I learned a while ago that e^(i*pi) is equal to -1, but I never knew why.  In a calculus textbook I was reading, he went off on a random tangent (not a sine (but now I might be going off on a cotangent mentioning this)) about imaginary exponents, and I realized what was going on while it was happening.

And now you know why all of those trigonometry formulae work too!

I hate teaching trig in classes that don't yet have Euler's formula.  Well, that's not true; I still enjoy trig.  But it is so much more elegant with Euler's formula.

Even without Euler's formula, I enjoyed trig because the teacher used the unit circle and explained, for instance, that the value of the tangent is called that because it's the length of an actual tangent line.  It blew my mind (and then, of course, Euler's formula blew my mind later.)

Edit:  Oh, and then the idea that both of these were related to AC circuits and that we can treat capacitance and inductance as imaginary resistance really blew my mind.

[Watno]

Can you elaborate on the tangent being the length of some tangent line?
I haven't heard of that and am quite intrigued to hear.

[SirPeebles]

Can you elaborate on the tangent being the length of some tangent line?
I haven't heard of that and am quite intrigued to hear.

Draw a unit circle and an angle (say, in the first quadrant) in the standard way, draw the tangent to the circle at the obvious point, and look at the length of the line segment joining the unit circle to the x-axis.  It is the tangent of the original angle.

edit:  by the way, it is just a one line proof by considering similar triangles, since the tangent is perpendicular to the radius.  Also, the length of segment joining the unit circle to the y-axis is the cotangent.

[GeoLib]

Can you elaborate on the tangent being the length of some tangent line?
I haven't heard of that and am quite intrigued to hear.

Draw a unit circle and an angle (say, in the first quadrant) in the standard way, draw the tangent to the circle at the obvious point, and look at the length of the line segment joining the unit circle to the x-axis.  It is the tangent of the original angle.

You can get the result by looking at similar triangles. Sin/Cos must be the same as that length/the radius of the circle (1).

[Kirian]

The tangent and cotangent are both segments tangent to the circle; the secant and cosecant are both segments secant to the circle.

Note this diagram also shows all the Pythagorean identities.

[GeoLib]

The tangent and cotangent are both segments tangent to the circle; the secant and cosecant are both segments secant to the circle.

Note this diagram also shows all the Pythagorean identities.

Wait, that isn't even the line segment I was talking about for tangent. The one from the point-of-interest to the x-axis tangent to the circle is also tangent, isn't it?

This is an awesome diagram though. I like how it shows all of the identities.

[sudgy]

The tangent and cotangent are both segments tangent to the circle; the secant and cosecant are both segments secant to the circle.

Note this diagram also shows all the Pythagorean identities.

Wait, that isn't even the line segment I was talking about for tangent. The one from the point-of-interest to the x-axis tangent to the circle is also tangent, isn't it?

The triangle made here and the triangle you describe are congruent, so yes.

And I agree, that's an amazing diagram.

[ashersky]

Did you try the xfrac package?

You made that word up, or stole it from Battlestar Galactica or something.  That's not a real think you can do to a package.

[Kirian]

Did you try the xfrac package?

You made that word up, or stole it from Battlestar Galactica or something.  That's not a real think you can do to a package.

My wife does that to my package all the time.

[qmech]

http://benjaminlmoore.wordpress.com/2014/03/06/most-common-rng-seeds-r-github/

Perfect for this thread:  A blog post I randomly found about the seeds programmers random use for their pseudo-random number generators.  (Spoiler:  Everyone just uses 1.  Seems legit.)

Aside:  In the far past when I've done programming and needed random, I usually seeded using microtime().  Anyone else do some variant of that?

I have a large book of BASIC program listings.  They typically start with the the enigmatic

Code: [Select]

RANDOMIZE TIME
and then, a few lines later

Code: [Select]

WHILE NOT FINISHED
(making use of the fact that undeclared variables default to 0.)

[SirPeebles]

Wait, that isn't even the line segment I was talking about for tangent. The one from the point-of-interest to the x-axis tangent to the circle is also tangent, isn't it?

This is an awesome diagram though. I like how it shows all of the identities.

You might prefer a coordinate-free description.  Just draw a unit circle (or any circle, and use the radius as your unit of measurement for lengths).  Then draw two rays coming out from the center where they meet at some angle.  Each of these rays meets the circle at one point; choose either one of them and draw the tangent line to the circle through that point.  The tangent line has a bounded segment between the two rays, and the length of this segment is the tangent of the angle.

Kirian's diagram and the one you envision differ only by which of the two points you drew the tangent line through.  They are of course equivalent by symmetry.

[WalrusMcFishSr]

This is the city of Masada, located in modern-day Israel:



See that massive earthen wedge there on the side of the cliff? The Romans BUILT that so they could assault the mountain fortress. Frikkin Romans, they're like ants!

[liopoil]

Just draw a unit circle (or any circle, and use the radius as your unit of measurement for lengths).

isn't that just the definition of the unit circle - that the radius is your unit of measurement?

[SirPeebles]

Just draw a unit circle (or any circle, and use the radius as your unit of measurement for lengths).

isn't that just the definition of the unit circle - that the radius is your unit of measurement?

More or less.  I just wanted to emphasize that the unit circle isn't some special circle; any circle is a unit circle with the right units.

[GeoLib]

Wait, that isn't even the line segment I was talking about for tangent. The one from the point-of-interest to the x-axis tangent to the circle is also tangent, isn't it?

This is an awesome diagram though. I like how it shows all of the identities.

You might prefer a coordinate-free description.  Just draw a unit circle (or any circle, and use the radius as your unit of measurement for lengths).  Then draw two rays coming out from the center where they meet at some angle.  Each of these rays meets the circle at one point; choose either one of them and draw the tangent line to the circle through that point.  The tangent line has a bounded segment between the two rays, and the length of this segment is the tangent of the angle.

Kirian's diagram and the one you envision differ only by which of the two points you drew the tangent line through.  They are of course equivalent by symmetry.

oh. duh. thanks!

[eHalcyon]

Science is scary.

More correlations here.

[Axxle]

I'm surprised so many people died by getting tangled in their bedsheets.

[SirPeebles]

Today I learned that vader is Dutch for father.  As in, Darth Vader.

[Axxle]

Must have made Empire Strikes Back a little less impressive in the Netherlands, then.

[mail-mi]

Today I learned that vader is Dutch for father.  As in, Darth Vader.

were you watching pitch perfect?

[SirPeebles]

Today I learned that vader is Dutch for father.  As in, Darth Vader.

were you watching pitch perfect?

Nope.  I don't know what that is.

[mail-mi]

Today I learned that vader is Dutch for father.  As in, Darth Vader.

were you watching pitch perfect?

Nope.  I don't know what that is.

a movie. they mention that fact in it.

[pacovf]

Science is scary.

More correlations here.

Some of these make a lot of sense:

Industrial Manufacturing Doctorates Awarded vs Per capita consumption of mozarella cheese
  -Those PhD students need paninis to survive

Per capita consumption of margarine vs Murders by bodily force
  -No one consumes margarine willingly, hence the desire to get rid of whoever is forcing you to do so

Aggregate worldwide income for petrolium companies vs Total revenue generated by golf courses
  -Only rich people play golf

Economic doctorates awarded vs People who died by falling out of their bed
  -Dirty conscience doesn't let you sleep well

Number of people killed by misusing a lawnmower vs Cost for 16oz of potato chips
  -Hypoglycemia

Worldwide non-commercial space launches vs Sociology doctorates awarded
  -Some people would rather leave the planet than having to listen to certain conversations

I notice that this site really likes marriages, bees, precipitations, and falling from places, but I don't see the correlation.

[Joseph2302]

I got my nails painted for the first time today