Is it cold outside? Why, it's positively polar.

01/31/2012

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It's time for another math blog!  This one shouldn't take too long, but I need to get it written before I can post about other things, so here it goes:  Polar Coordinates!  Up until now we've dealt exclusively with Cartesian coordinates when plotting points on a graph (Cartesian coordinates are the ones you learn in high school geometry), and they look like this:
Pretty simple.  (3,2) accurately describes this point because its value on the x-axis is 3, and its value on the y-axis is 2.  Now, if this were a map and I was telling you how to get to (3,2) from (0,0), I'd tell you to walk 3 units east and 2 units north.  That's essentially what Cartesian coordinates do.  Polar coordinates, on the other hand tell you how to get to a point as the crow flies, to continue the map comparison.  Thus:
So how do we define a set of coordinates that will direct us along our red line to end up at (3,2)?  Do get there from (0,0) as the crow flies, we need two pieces of information:  a heading, and a disatance.  And that's what polar coordinates are:  an angle, expressed in radians (we'll call it θ), and a distance, which is the length of our red line (we'll call this r, for radius), and it's written as (r,θ).

Pretty simple, yes?  By now it should be clear how to find these values given what we already know.  We'll start by finding c using the trusty Pythagorean theorem:

a^2 + b^2 = r^2
2^2 + 3^2 = r^2
4 + 9 = r^2
13 = r^2
√13 = r
So there's one of our two coordinates.  Let's find the value of the angle θ now, shall we?  Seems the easiest way, or at least the way that involves all integers, is to find the arctangent of 2/3.  Plunking tan^-1(2/3) into our trusty calculator in radian mode, we see that the arctangent of 2/3 is approximately 0.59 radians, which coverts nicely to 33.69°, so our polar coordinates would be expressed (√13,33.69°).  Thus:
Now, that was easy enough, but arctan only works in quadrants one and four (in other words, angles whose cosines are positive).  There's a whole slew of functions to allow for that, and to save all those penstrokes, it's usually expressed as atan2 (y,x).  It's a simple way to express ALL THIS:
Which is a complicated way to say "add or subtract 90 or 180 degrees as necessary to figure this out in quadrant one or four". 

Now, converting from polar coordinates to cartesian coordinates is pretty simple too, and I won't go through the process of proving these formulae, but here they are:
x=r cosθ
y=r sinθ
Let's work it out given our same problem from above.x=√13 cos(.59)We rounded that angle to the nearest hundredth, so our answer is a little off, but plugging that in a calculator does give us approximately 3.  Similarly,
y=√13 sin(.59)
Yields a decimal that's pretty close to 2.  So - our calculations were a success!  

That's all for now - soon we begin writing about pre-calc, so stay tuned for that.
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Tan to the negative one, it turns out, doesn't mean you're very pale.

01/12/2012

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I have returned!  Well, in truth, I never left.  I've just been working, studying, and otherwise not writing, and yesterday I realized that I've gotten so far ahead in studying math that it's going to take several posts to catch up!  That's not a bad thing - so I'm going to embark on a whirlwind tour of some of the things I've learned since our last meaningful time together in November (wow - has it been that long?), including some new trig concepts like polar coordinates, some neat problems I've run across, and even some brief forays into the mystical land of calculus!  This first post, though, will just tackle the idea of the Inverse Trig Functions.  Here we go!


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Inverse Functions
So we know about sine, cosine, and tangent... and we even know about cosecant, secant, and cotangent, their multiplicative inverses.  So these inverse functions are the compositional inverse.  I'm talking about arcsin, arccosine, and arctangent - but if you've seen a graphing calculator, you'd probably recognize them by their more common notation: sin^-1, cos^-1, and tan^-1.  Don't let the negative one exponent fool you - this is simply how we denote that we're looking for the inverse function.  So what is that anyway?  Well, look at the triangle here.

Say we know the length of sides a and b but we don't know the value of the angle θ. 

Given what we already know, we can clearly say that tanθ = a / b,  but that doesn't really tell us what the value of θ is.  That's where the inverse functions come in.  See, where the tanθ = a / b, the arctan a / b = θ!  The same is true for the sine, cosine and all the rest.  Simply put, when you're trying to get an angle and you know the values of some sides, you're going to be looking for inverse functions.  Plus - they can be graphed just like our other functions can, but the results may surprise you.  Take a look at what happens when we graph arcsin(x):
The graph if sin(x) is in red - and the graph of arcsin(x) is in blue.  It does something pretty interesting, and if we look at the green vertical line we can see that unlike the sine graph, the arcsine graph is not a function.  

Now - think about this for a second and you may realize why this is the case.  Picture the value of arcsin(x) as the angle at the center of the unit circle (because that's what it is).  That value ranges, in each "quadrant" of the graph, between  π/2 radians and - π/2 radians - or, 90 degrees and -90 degrees.  Now picture those values rolled out as you continue around the circle.  Your angle may be 5π radians, but you're in the same place as if you had only gone  π radians - remember, one trip around the circle is 2π, so reduce in terms of that.  Also, see that the value of sin(π) is the same as the value of sin(0).  They're both zero.

In order to keep this from happening with inverse trig functions, and to actually make them functions, as the name implies, we have to define ranges for them.  If we define the range of arcsin(x) as [- π/x,  π/2] and the domain as [-1,1] we get something much nicer:
Plus, if we hit it with a vertical line test, we can see that yes, it indeed is a function now, and all is right with the world.  This is the part of trig where one has to start thinking a little more aesthetically about the way things work - but after a while, it just becomes intuitive.  There are a lot deeper implications to the inverse functions than I'm going to get into here, but this is a good introduction to the concept, I think.

That's all for now - more to come very soon, promise!
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Please Excuse My December Article Schedule

12/20/2011

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I know I'm way past due for another article and I do have plenty to write about: polar coordinates, using trig in navigational situations and other real-world applications, but it's December and the holiday schedule, between work, studying, and preparing for festivities, just don't leave a great deal of time for writing.

I promise I'll be back in January with more maths, but until then, Please Excuse My December Article Schedule.
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The Law of Cosines, Law of Sines, and a Quadratic Surprise

12/03/2011

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So it's past time for another math post.  Don't worry, my studies are continuing, but I've discovered that distilling all the great mathy things I'm learning into a coherent blog post is actually more difficult than studying the math!  Still, that's the reason for keeping the blog - if I can't explain what I'm learning in simple words, I don't truly understand it.  

So - it's taken me a little while to prepare a post on the Law of Cosines.  Mostly because it took me a little while to wrap my brain around it - but a little bit of lightning struck my brain and I suddenly got it.  Here's the gist:
Here's a triangle.   It has sides of length a, b, and c, and angles which measure A, B, and C.
Now - note that there are no right angles here, which makes this triangle different from any of the other ones we've used so far.  The law of cosines and the law of sines are used to determine the relationships between the angles and sides of these triangles, and frankly I think the law of cosines is the more useful of the two, but I'll show you both.

Now, for a right triangle we can calculate the relationship between its sides pretty simply, using the Pythagorean theorem.  The Law of Cosines, in fact, is just an extension of that which will work for any triangle.  Here it is:
Now.  I have been through the proof for this equation a couple of times, and while I appreciate it, I sort of resolved just to smile and know that this is true.  The lightning-bolt moment came when I tried to apply the law of cosines to a right triangle, where C=90 degrees.  The cosine of any 90 degree angle is the same as the x coordinate of that radius' endpoint on the unit circle, remember - so, it's zero.  Plug that into the equation above and it becomes, quite beautifully, the Pythagorean theorem. 

It also works for finding the length of any of the three sides, not just c, as long as you replace the angle after "cos" with the angle opposite the side you're trying to find.

In Practice

So let's give this a shot.  Back to our original triangle, let's assign some values.
a = 8
c = 11
B = 37 degrees.
So we know the lengths of two sides, and the measure of one angle.  Let's see if we can use that to find out the length of side B.  We'll plug the values we know into the Law of Cosines (with apologies for the lack of superscript, I just can't be bothered to create images or html for all this crap.  :P

b^2 = a^2 + c^2 - 2ab cos(B)

b^2 = 8^2 + 11^2 - 2 8 11 cos(37 deg)

b^2 = 64 + 121 - 176 cos(37deg)
b^2 = 185 - 176 cos(37deg)

We'll have to use a calculator to find the cosine of 37 degrees, or look it up.  Here it is.

cos (37 degrees) = 0.798...

so...
b^2 = 185 - 176 (0.798...)

b^2 = 185 - 140.559...

b^2 = 44.440...

b = sqrt(44.440...)

b = 6.666...

Wow.  I promise I didn't plan that.

I did use a problem that I had the answer to, just to verify that I did it correctly but I didn't look at the solution until I was done.  Success!  Now we can find the other two angles, if we want.  If you're looking for the angle, you just have to reconfigure the equation solving for cos(x). 

Thus:

c^2 = a^2 + b^2 - 2ab cos(C)

c^2 - a^2 - b^2 = -2ab cos(C)

-c^2 + a^2 + b^2 = 2ab cos(C)

(-c^2 + a^2 + b^2) / 2ab = cos (C)

There you have it.  Yay algebra!  Now, here's the less useful Law of Sines.

The Law of Sines

The Law of SInes is simply a statement of the ratios between the angles and their opposite sides.  So, in our same triangle from before:
Simply put, the ratio between the length of side a and the sine of A is always the same as the ratio of the length of side b and the sine of B, and so on.  Useful to know, but I gather the Law of Cosines is used more often.  I've been through the proof of this rule as well - I think it's useful to understand why these things are true and not just accept that they are.

The Quadratic Surprise

I had forgotten completely about the quadratic formula.  Imagine my surprise when I stumbled upon a trig problem that seemed just unsolveable at first because I couldn't isolate the variable.... when I looked up the solution, I saw that you had to rephrase the equation as a quadratic first.  I don't remember the details of the problem but a quadratic equation, basically, looks like this:

ax^2 + bx + c = 0

Where a, b, and c are given, and are known as the quadratic coefficients.  Then you use the quadratic formula to solve for x:
From there, just plug in and solve.  I would never have gotten here by myself, though, because I didn't immediately recognize the problem as quadratic.  When I first started this endeavor, someone told me that when they get to calculus, most students' major handicap isn't geometry or trigonometry, it's algebra.  I can see here why.  The Quadratic Formula was something we talked about in Algebra 2 (I think?) that I hadn't even though about since.  I'm pretty sure I'll have to go back and review algebraic ideas like this from time to time.  Is it strange that, rather than getting frustrated, I regarded it as a nice surprise?  Probably is... but I am having fun, and the major upshot of teaching myself this stuff is that I have time go back and review older ideas like this, because I don't have to stick to a schedule, and I'm not depending upon a grade.  I'm very thankful for that!

Until next time...
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Thankfulness

11/23/2011

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No math today, but I promise I'll be writing about irrationals, transcendentals, and the law of cosines soon.  Today I want to share something that I am thankful for.  It's a photograph.  Those of you who know me probably know the photograph I'm referring to - there's a framed print next to my desk.  The first time I saw this image, I was fundamentally changed.  I have always been interested in science and astronomy, but my interests have been purely philosophical and had very little to do with the math.  Seeing this image, realizing its implications, made something in my head 'snap.'  This image continues to this day to be the most humbling and awe-inspiring photograph I have ever seen.  

First, a little history.
In 1995, astronomers did something with the Hubble space telescope that, on the surface, may seem sort of odd.  They pointed it at a small, utterly empty area of space.  The area, in the constellation Ursa Major, is no larger than a grain of sand held at arm's length to those of us on earth.  This is what it looks like.
See?  Unremarkable.  The moon, by the way, would fill about a quarter of this image if it were present.  So - a very, very small, very, very empty piece of sky.  So why do this?  Well, the same reason that astronomers prefer to be in the middle of nowhere when they observe the sky - light pollution.  We wanted to see what was out there, past what we can see now.  If anything.

For ten consecutive days, the Hubble gazed into the abyss.

This is what it saw.
Yeah.

This is the Hubble Deep Field.  Just so we're clear, there are a handful of local stars in this photograph.  The bright spots with cross-style lens flares, those are stars in our galaxy.  There aren't many, just four or so.  Everything else - every spot, speck, smear, pinpoint and mote of dust - is an entire galaxy.  All told there are three thousand galaxies in this image.  

Three thousand galaxies - in a speck of empty space.  

But the story doesn't end there, not quite.

My god, it's full of stars...

In 2003, we did it again.  This time, the Hubble had improved optics, and we chose an tiny, more remote patch of sky and looked at it over a longer period.  Between September 2003 and January 2004 the Hubble stared at a miniscule patch of sky in the constellation Fornax (near Orion).  It's a piece of sky that is roughly one-seventieth of the area subtended by the moon in our sky.  This image, the Hubble Ultra-Deep Field, is the most important photograph ever captured by humans.  It is a picture of 13 billion years.  This is the photo that hangs by my desk.  

This is what it looks like.
There are ten thousand galaxies in this photograph.  

Some of the tiniest ones, shifted far into the red, are images from just 400 million years after the Big Bang.  Light from these distant, young galaxies ended their 13 billion light-year journey a speck, captured literally one photon at a time, on Hubble's collector as it gazed into nothingness.

This is a picture that inspires and humbles me on a very intense, personal level.  It inspired me to stop thinking of science philosophically and strive to learn more.  To truly comprehend.  To be greater than I am.  Because in the end, we're just an idea.  A mere wisp of thought tucked deep inside a speck of light - perhaps collecting, one photon at a time, on the collector of another optical device, billions of years from now.

One of an infinite number of angels, dancing on the head of a distant, ancient pin.

But now I'm getting philosophical again. 

So.  Of course I am eternally thankful for friends, and for family.  I'm eternally thankful to have a wonderful partner to dance on this tiny world with.  But in the larger sense, I'm thankful for the Hubble for giving us an eye to peer into the void.  I'm thankful for these two images and for .  I'm thankful for the universe, which is a more amazing and mind-boggling thing than any of us can truly comprehend, and finally, I'm thankful for perspective.

Happy Thanksgiving, everyone.
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The quest for identity

11/20/2011

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Identities!  That's been the crux of my studies of late and I'm sure this won't be the last time I visit them.  I took a snapshot of my handy-dandy notebook to the left - that page is full of trig identities, and that's just a few of them.  There are countless identities that can be derived, and that's pretty fascinating honestly!  

An identity is a mathematical statement that is tautologically true.  That is, it is true no matter what numerical value is assigned to it.

Here's an example of a simple one:

tanɵ = sinɵ/cosɵ.

Since we know from graphing some functions that the sine of an angle is equal to its y coordinate on the unit circle, and the cosine of an angle is equal to its x coordinate on the unit circle, we can then derive that our original formula tanɵ = opposite / adjacent can also be stated tanɵ = sinɵ/cosɵ, which is honestly a pretty elegant way to relating our three major trig functions to each other.

There are three other trig functions that I've been neglecting mentioning and I guess here is as good a place as any to introduce them.  These are the inverse functions:  cosecant, secant, and cotangent.  They're just the multiplicative inverses of sine, cosine, and tangent - respectively.  Thus:

cscɵ = 1/sinɵ
secɵ = 1/cosɵ
cotɵ = 1/tanɵ ... or ... cosɵ/sinɵ

See how that works?  You just invert them.  Also note that the prefix co- is present in the inverses of only the functions that don't already have one:  sine & cosecant (abbreviated csc), cosine & secant (abbreviated sec), and of course tangent  & cotangent (abbreviated cot).  These are the most basic trig identities.  These statements are true no matter what the value of ɵ is.

I'm not going to go through proofs for a bunch of identities here (not in this post anyway - I may do one or two later if I feel like it'll help my understanding) - there are resources that will help you with that  if you want to pursue them.  But I will give you another basic one, and one that pretty much blew my mind.  
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The Pythagorean Identity

The name will probably be familiar to you, and for good reason.  To the right you can see a lovely graph laying it out, but I'll take you through the proof.  It's pretty elegant really, the way it plays out.


We start with the Pythagorean Theorem:
a2 + b2 = c2
Now, apply this to the values that we know from our studies already.  Given the unit circle definition, we can say that that c=1 because c is the hypotenuse, right?  And we can also substitute sinɵ and cosɵ for a and b.  Which gives us this:
sin2ɵ + cos2ɵ = 12

Which is to say...

sin2ɵ + cos2ɵ = 1
And there you have it.  The first Pythagorean identity.  It doesn't look like much until you remember that this is always true, no matter what the value of ɵ is.  It pretty much blew my mind.  We can derive other Pythagorean identities from this first one.  Let's begin by seeing what happens if we remove the square of the cosine.  To do this, we divide both sides by cos^2 ɵ. 
sin2ɵ   + cos2ɵ   = 12
cos2ɵ   + cos2ɵ = cos2
Some of those things we've seen before, in this very post.  sinɵ/cosɵ is the same as tanɵ.  It follows that if we square the whole thing, like we have here, we get tan^2(ɵ).  The second item, cos^2(ɵ) / cos^2(ɵ) equates out to one (because any number divided by itself is one).  And finally, 1/cosɵ is one of our new functions:  secɵ.  Again, if we square all of that as it is here, sec^2(ɵ).  So:
tan2ɵ + 1 = sec2ɵ
Neat, huh?  That's another identity.  It's always true.  We can keep deriving more identities this way by playing with the way the equation is put together.  Identities are pretty brain-bending at first, which is part of the reason this post took so long to prepare - but once you wrap your brain around them, the relationships between the different parts of everything starts to take on a new light.

And finally, I'll leave you with the solution to last week's trivia.
When last we met, I asked you why the tangent wave approaches infinity at certain points.  Well, let's look at how we find the value of tanɵ again:

tanɵ = sinɵ/cosɵ.
Now remember, the sine of any angle is equal to its y coordinate on the unit circle.  The cosine is equal to its x coordinate on the unit circle.  Now look at where the wave approaches infinity, and find those spots on the unit circle:
Remember we're dividing y by x to find the tangent - what's the x value of π/2 and 3π/2?  In both cases, zero.  And what happens when you divide by zero?  

Illuminating, Isn't it?


To infinity and beyond!  See you next time.
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The functional weekend continued

11/12/2011

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Welcome back!  When last we met we had just graphed a sine function - today we'll look at the anatomy of the sine wave and how to express it in a simple equation.  The example above is the sine wave I graphed yesterday, and its equation is below it:

f(x) = sin(x)

Straightforward enough, but let's take a look at why.  The equation of any sine function is written like this:

f(x) = A sin (P x)

Where A is the wave's amplitude, and P is the period.  The period of the function is how many revolutions it completes in  2π radians. This got a little confusing for me, because no two textbooks seem to be able to agree on variables for the period or how you find it, but you can think of it this way - look for the point where our wave starts to repeat, and call that value |B| (that's the absolute value of B - so treat it as a positive number even if  it's negative). Then, simply solve for 2π/|B|.  This wave, since it's a sine wave, starts at zero, rises to its maximum, goes back through zero, into the negative, and then arrives at zero again at exactly 2π.  So, given our formula, 2π/2π = 1, so the period is 1.  

In the above example, the amplitude is also one (since the wave's range is between one and negative one - the measure of the amplitude is always half of the wave's total height on the graph - in this case, only the distance between the horizontal axis and the maximum value.   

So, with an amplitude of one and a period of one, f(x) = 1sin(1x) = sin(x).  Easy?  Okay, let's look at a cosine wave.
I pulled this, and the rest of the graphs in this post from the Khan Academy's modules on graphing trig functions, so if you want to try your hand, head over there and get your learnin' on.

So, here's a cosine wave.  It's a little different from the sine, in that whereas the sine wave starts at zero, the cosine wave begins at its maximum value, so when you're looking for its period, you'll look for the point where it reaches its maximum again.  The wave above, for instance, still has a period of one, since it completes a revolution at 2π.  The amplitude, however, is different.  Its maximum is at 2.5, so that's the amplitude - now, we like to express things in our function equations as fractions rather than decimals, and 2.5 as a fraction is 5/2 - five halves.  So this graph would be expressed thus:

f(x) = (5/2)cos (x)

Let's look at a couple more examples.
Okay, so when we look at this, we can immediately see that it's a sine wave, and a non-shifted one at that (sometimes they will be shifted to the left or right of zero - the distance from zero is called the phase of the wave function).  We can also see that the amplitude is 1.5, or 3/2 as a fraction.  The wave begins to repeat at 1π, so if we solve for 2π/1π, we get 2, so the period is 2. 

f(x) = (3/2)sin (2x)

Make sense?  Here's another.
Okay - this is a cosine wave, clearly, since it begins at its maximum.  The amplitude is one, since the maximum is one, but let's find the period.  This wave repeats at 6π, so 2π/6π = 1/3.   Thus:

f(x) = cos (1/3)x

Not too difficult, eh?  I'll leave you with something fascinating to ponder - one wave function we haven't looked at yet is the tangent function.  It does something interesting.  Take a look:
Weird, huh?  I'll leave you to ponder exactly why the tangent wave does this - but here's a hint.  At 90 degrees (π/2 radians) and 270 degrees (3π/2 radians), the value of tan ɵ actually approaches infinity - that's why it spikes off the grid like that.  If you can noodle out why that happens, leave a comment.  I'll give you the answer in the next post. 

Have a great weekend everybody!
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Function Friday

11/11/2011

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Hi everybody, and welcome to Friday.  Today's post deals with the next step in my trig studies - graphic trigonometric functions.  Truth be told I'm actually studying identities now, but I need to write about functions before I can really get into those.  So - what is a trig fuction, and how do we graph it?  Mr. Peabody, set the Wayback Machine for Geometry class!

For those of you who have taken geometry, you may remember functions vaguely.   Functions deal with the way two sets of numbers relate to each other - let's look at slope-intercept form equations as a simple example.  We wanted to graph the line expressed by the equation y=2x+3, what would that line look like?  Well, to find the answer, look at the equation.  For every value of y, we can calculate the corresponding x coordinate.  We do this by making a little table:
yx
15
27
39
411
513
So we've just plunked in values for y - in this case, 1 - 5, but it doesn't really matter which numbers you use.  And then we've solved for x.  When y=1, x=5 because 2(1)+3 = 2+3 = 5.  When y=2, x=7, because 2(2)+3 = 4+3 = 7.  And so on down the line.  You can do the negative values of y as well, if you want, and many times you'll get a lovely parabola, but in this case the line's slope is constant even into the negative.  Here's what it looks like when we graph that:
So there we see that for every value of x, there is exactly one value of y, and therefore we can say that y is a function of x.  The inverse is also true.  Now - let's look at our unit circle again.
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This is where graphing functions get interesting, and how they're used in real life.  You graph functions in terms of the relation between two things - the number of customers in a store over time, for example, or as a more depressing example, the invariably downward trend of the amount of money in the bank as a function of time.

Now, look at our unit circle.  When we graph trig functions we're going to graph all the values of either x or y as a function of radians.  Why do we want to do this?  Well, maybe you'll see in a minute.  

The first thing you'll notice is that the values of either axis,  y for example, have a range of -1 > y > 1.   That means that y is never less than negative one, and never greater than one.  Easy enough - but now let's see how we graph the values of y as a function of radians.  We'll make another table:
rady
00
π/21
π0
3π/2-1
2π0
Make sense?  Look at the circle again if you want.  Of course you could go on and calculate negative values, or values higher than 2π, since you'd just start around the circle again, but for our purposes this is all we need.  Now let us see how that looks graphed as a function.
Behold the sine wave.
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Why is it called a sine wave?  Well, here's something interesting.  These are values of y for our unit circle, but let's put a point on one in particular, and draw a radius to it.  We'll use π/4 for simplicity's sake.  The radius intersects our circle at a point P, which is at a location we don't really know - (x,y)

Now - how can we learn the y coordinate of the point P?

Well, we can figure it out without too much trouble, actually.

How?  Well, let's start by making a triangle out of the pieces we already have:
Now that we have a triangle to work with, we can go back to SOH CAH TOA.  We know that the angle at the center of the circle is π/4 radians, and since this is a unit circle we also know the length of the hypotenuse - it's 1.   So, if we're looking for the y coordinate, we're looking for a point on the y axis that is at the same height as the opposite side of our triangle.  So, we simply need to determine the length of the opposite side, and that number will be our value for the y coordinate of P.  In other words, we're looking for the sine:  sine = opposite / hypotenuse. 

sin(π/4) = y/1
Wow - look what just happened.  We can reduce that to...
sin(π/4) = y
Yup, the cosine and the y coordinate are the same thing.  That's true for any angle.  

Similarly, the sine and the x coordinate are the same thing.  So, we have two new rules:
cosɵ = x
sinɵ = y

So, the coordinates of our point expressed as an ordered pair (x,y) are ( cos(π/4),sin(π/4) ), which makes sin(π/4) our y-coordinate, and we find that it works out to y = √2/2.  

We would find that answer by either looking it up or just knowing it - in this case it's the latter, since we're dealing with a 45 degree angle.  Incidentally, the cosine of π/4 is also √2/2, so the x coordinate would be the same as the y - in a 45-45-90 triangle that's always the case.  
√2, incidentally, is quite the fascinating number in and of itself, and its discovery is due to the 45-45-90 triangle, and will this be the subject of my next post on irrationals, but that's a topic for another day.

There's quite a bit more to graphing trig functions as well, but I'll let you digest that much for now, and over the weekend I'll put together a follow-up that gets into cosine functions, tangent functions, their anatomy in terms of period and amplitude, and more mathy goodness!  Thanks for reading as always!
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My Love Affair with Irrationals, Part One: Pi

11/06/2011

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 Here it is, the long-promised irrationals meditation.  I've long been fascinated with irrational numbers, but not until I started studying trigonometry in depth did I come to realize (or, rather appreciate) a fact that was a little startling to me at first - the overwhelming majority of numbers in mathematics are, in fact, irrational.  It's not a surprising fact, given what I know about numbers - but it was still a startling realization to see in black and white in my studies.  

Well - before I get too esoteric, let's start by deciding what an irrational number really is.  Follow me down the rabbit-hole...

The types of numbersIt may be good to begin by describing the types of numbers found in mathematics - don't worry, this isn't going to get too mathy, but it will help illustrate where irrationals fit into the scheme of things and why it's so amazing to me that they vastly outnumber every other type of number in mathematics.

Natural Numbers:  These are the numbers that we count with.  1, 2, 3, 4, 5.... they don't include zero, decimals, or any negative value.

Whole Numbers:  Whole numbers include all the natural numbers, plus zero.  0, 1, 2, 3, 4, 5...

Integers:  Integers include all the whole numbers, plus all the negative numbers.  Still no decimals:  ...-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5...

Rational Numbers:  Rational numbers include all the numbers - and this may get a little brainy, but bear with me - that can be expressed as a fraction a/b, where a and b are both integers.  In other words, rational numbers include all the integers, all decimals that terminate (such as 0.25, which can be expressed as 1/4), and all non-terminating decimals which repeat.  For example:  0.333... is a non-terminating decimal, but since it can be expressed as a fraction 1/3, it is said to be rational.   Similarly, a decimal like 0.142857142857... may look irrational, but note the repeating set of numbers 142857.  This, too is a rational number (1/7).  So, in effect, all integers are rational, all simple fractions are rational, and all decimals that either repeat or terminate are also rational.

Irrational Numbers:  It follows, then, that the irrational numbers would be any number that is not rational.  That is - any decimal that goes on forever without ever repeating or terminating is said to be irrational.   They fascinate me because their true value can never, ever be known - we can calculate pi out to several million decimal places, but there are several million more taunting us, reaching off into infinity.  

Irrational numbers are the spaces between the numbers that we're comfortable with - they're hidden, they're unique and they're un-knowable - and they outnumber everything else.  

This is no longer math to me.  

This is poetry.

For the sake of being complete, I'll note that the group of all rational and irrational numbers, both positive and negative, make up the Real Numbers, which leaves only Imaginary Numbers and Complex Numbers, which are also fascinating to me, but that's a subject for another post.    Let's get back to the poetry of irrational numbers.
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pi

We'll begin with one that everyone knows:  pi.   The value of pi, though unknowable, is well known:  3.1415926... off into infinity.  Pi is defined as the ratio of a circle's diameter to its circumference, and is used all over mathematics, particularly in trig.  I've spent a whole lot of time with one particular iteration of pi - 2π.  We all know that there are 2π rad in one circle, and the term 2π is used all over the place in math.  Now - this actually presents a minor inconvenience, because essentially you're dealing with one unit circle that adds up to two rather than one - all over trigonometry.  When calculating the period of a sine or cosine wave, you must determine how many times that wave repeats in 2π radians, not one.  2π is everywhere.  Now this isn't too bad once you get used to one, but some mathematicians are proposing an alternative to 2π, and a new irrational defined by a greek letter:  tau.
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Tau or τ, is shown to the left.  Its value is simple.  τ = 2π.  Its implementation leads to some very elegant results when we apply it to what we've already learned in trigonometry.  Observe what happens when we apply tau to our unit circle, with some graphics courtesy of tauday.com, which you should definitely read if you have any interest at all in this stuff:
This is a unit circle, again - divided into fractions of a circle.  180 degrees, of course is half a circle, 270 degrees is 3/4 of a circle, 45 degrees is 1/8 of a circle, and so on.  Pretty straightforward.  Now look at these same measurements expressed in radians, in terms of pi.  
Now we've covered this before here, so I won't spend a lot of time on it.  But as you see, our fractions aren't quite so intuitive when one whole circle adds up the two pi radians.  1/2 of 2π is 90 degrees, not 180.  Now look what happens to these numbers when we substitute tau for 2π.
That,  dear readers, is a thing of beauty.  When we substitute tau for 2π, something amazing happens - our original fractions appear, intuitive and elegant.  One half of a circle is one half of tau radians.  One eights is one eighth tau radians.  When I first read this, it blew my mind.  The implications keep going, so please take a look at tauday.com and allow the glory of tau to overwhelm and amaze you as it did me.

I had meant to cover several more significant irrational numbers in this post, but I've run out of time, so I'm going to make this a series of posts instead.  Stay tuned for more dissections and explorations on irrationality in part two...
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An unexpected close encounter

11/04/2011

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Tonight I fully expected to write up a lengthy and probably needlessly esoteric tome on my fascination with irrationals, which I still plan on doing, but now I have something a little more exciting to start with.   Tonight as I parked and got out of my car I caught sight of a pair of especially bright spots in the sky, and that threw me for a second.  I knew that one of them must be Jupiter, but the second one was a mystery.  Saturn would be a candidate, since it's nearly as bright as Jupiter, but it shouldn't be anywhere near Jupiter in the sky right now - then I noticed it was moving.  Fast.  It was then that I figured out what I was looking at.  

A CLOSE ENCOUNTER!  Yes, I had one... but this one wasn't with strange visitors from another planet, the pilots were 100% earthling.  I paid a visit to Heavens Above just to confirm, and within seconds my suspicions were validated.  Observe:
That's the orbital path of the International Space Station - and it's the culprit all right.  The current position and orbital path of the ISS placed it in exactly the right position for me to see it fly by.  I checked the details for my location just to be sure, and it was confirmed:  an ISS flyover at 7:42 PM.  Nice!

You can use Heavens Above to schedule your own close encounter with the ISS, which is easily visible as it passes over - it's very bright - or maybe one evening you'll be lucky enough to have a surprise encounter like I did.

So if you see a really bright star-like point moving at a breakneck pace (and it usually fades in brightness as it passes over, so it could appear to fade in and fade out, very myterious and UFO-like) take a peek at Heavens Above before you go calling in the team from Area 51.  There are plenty of man-made satellites in orbit that will catch the sun's light just as the moon and planets do, and many of them shine brightly enough to be frequently mistaken for little green men.

More math coming soon, I promise!  It's a crazy vacation weekend and I've had barely any time to calculate anything that isn't my bank balance, and that's far more intimidating than trig.  Yeesh.
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