Pathagorean Identities

[Lydia_R]

While I was treking from Montana to Oregon this summer, I stopped at my old stomping grounds, the downtown Seattle library. The internet was down there and there were very few people in the building. I had a good browse in the political history sections and then bumped into these Pathagorean Identities in the math section.

 

I got a new phone last week and this is one of the first pictures I took with it. I love the resolution and content of it. 

 

The idea here is that if you have one of the trigonometric functions, you can derive the others. In this case I am using cosine to solve Sine. Not sure what else can be done with these.

 

I got interested in reciprocals a few months ago. The sine and cosine of 45 degrees is .707. The tangent of 45 degrees is .707/.707 which equals 1. If the division of y/x is greater than 1, then the angle is greater than 45 degrees and the reciprocal is less than one and less than 45 degrees.

 

I'm just studying this stuff as a hobby. I keep on thinking about reducing a fraction to a decimal and then converting it back to a fraction. A y/x relationship/ratio. I was using ratios extensively in coding a video game collision detection in 2003 before starting to use trigonometry in it in 2004. It's interesting the relationship of ratios in trigonometry.

 

Reciprocals are the 1/x function. You can type in a number then hit the 1/x to see the reciprocal and then hit the button again to see it flip back. Interesting stuff if you ask me!

[Lydia_R]

And this all leads me to the idea that if I take any number greater than one and do the 1/x reciprocal of it, the result will be less than one. And then I can replace that cos(pi/6) with that number to find x which is theoretically sin(x).  And I can then reverse those two numbers to express the complimentary angle (it would be the same arc distance on the other side of 45 degrees).

 

My theory is that by doing that....  Well, it's weird. I'm taking one single number in space and turning it into an angle which can be expressed in radians, degrees or as rise/run.

 

Just kind of playing with math here. I just downloaded this nice calctastic app and ran some calculations. This .523 is one I memorized when I was learning trig 20 years ago. Circumference measure to 30 degrees.  The .577 number is the one I'm trying to figure out or work with. I suppose I should go back to paper and pen now.

[Lydia_R]

Are you one? Identify....

[Charlize]

i'm definitely a squared one when dealing with math.  No ? (question) of that but saying that i'll be calculating  the amount of copper tubing for a solar hot water heat exchanger lather the morning.

 

Hugs,

 

Charlize

[KathyLauren]

I always was a math nerd.  I took a photo last month that included a meteor.  Someone else, about 60 km away, took a photo of what I was pretty sure included the same meteor.  After asking him for his latitude and longitude (which he was reluctant to tell me, since he likes to keep his favourite photography sites secret), I was able to calculate where in three dimensions the meteor was.  (It was 51 km from me, 99 km from him, at a height of 75 km.)

[Mmindy]

I was never very good or confident with math. I always praise the teachers and my mother for working so hard to see that I passed freshman algebra in my senior year. It was the only credit I needed to graduate. I was always asking what is X and why am I dividing or multiplying it by Y? What is Y? Everyone said it could be anything, but not give examples or try to explain it in practical ways. I couldn’t get past that. It was years later, and I started taking trade school classes through the Union, to acquire a first class stationary engineer license. I learned to apply math and algebra to those needs quicker than four years of high school. Why? Because the instructors were skilled in the practical applications before moving on to the theoretical. I wish those teaching methods could be used at a younger introductory level. Now I look at mathematical formulas with a different level of understanding, and I do have access to a calculator/computer every waking moment of my day. That leads me to this.

 

 

[Lydia_R]

I love this calculator app.  I've noticed how it puts the "r", "d" or "g" above the trigonometric functions.  I essentially do the same thing in my programming work.  I typically do this:

 

cos(30 * dg);

 

I define constants for dg and rd at the top of my script (degrees [pi / 180] and radians [180 / pi]).  Then when I use 30 degrees like I did in my example, I multiply it by my dg constant to translate it to radians because the cos() function only takes radians.  And when I do something like an atan2() (arc tangent), then I typically multiply it by my rd constant to move it back into the degrees space.  I just find it easier to do my math in degrees so I can do something like (angleX - 30) instead of (angleX - (pi / 6)).  The older I get, the more comfortable I become at thinking and working with radians, but 20 years ago, it was way easier to think in degrees.

 

This calculator app just looks so fantastic on my phone.  I set it down on a table and the whole thing just looks like a calculator straight out of the 1980's.  I really haven't watched any significant traditional TV since the 1980's, but I'll sit at a table these days and just plug numbers into this calculator for 30 minutes.

 

Height of 75km.  Wow!  hehe, the squared one!  I was just kind of flipping out when I saw those Pythagorean Identities and the concept of 1 squared!  What a weird reality!  It's like a lot of the math I'm doing lately with dividing a number less than one by a number less than one or a number less than one divided by a negative number less than one....  I work most of those things out on paper, but I really don't know the rules well enough to do all that math accurately.  And then I find myself pondering how even most clocks with hands these days are digital devices.  And how the biggest circle I can get my hands on easily is a bicycle wheel.  Well, thinking now, I guess I could get a hula hoop!

 

I remember watching a bumble bee in a park 21 years ago and observing how it deals with gravity.  And thinking about how a piece of Styrofoam can have magnetic properties.  I keep on coming to the idea that all creatures are basically seeing the same world no matter what size they are.  They see from different places according to what their size allows them to get to, but it is the same kind of picture unless their eyes are distorting it.  I often observe how a bird can just be sitting there and then all the sudden be moving incredibly fast away from the scene.

 

I tend to focus way more on the physical environment than social issues.  The concept that I've come up with this summer is that I could have a wonderful partner in a garden of eden type place, but I would still want some kind of instrument to play with.  The question comes up on whether having the thing/device/instrument to play with is enough without the partner?  It seems life is somewhat lacking without both things.  I tend to be pretty happy with objects though.   Smashing all hope of a potential future partner, being totally isolated without communication, stranded on Mars without communication and knowing it will never be re-established.  Things like that tend to go through my mind.

 

I seem to have gone through this thing the last year of constant downsizing and moving my things around with the sale of my house.  What am I keeping?  What can I sell?  For how much?  Why don't people want this wonderful thing that I have that I don't want to get rid of but have to?  Wondering around a thousand miles from home with no money and just the clothes on my back and some hygiene supplies.  A full can of pink shaving cream in my pocket!  The bare essentials of this transwoman!  At one point I had a fancy hand bag with me with large bottles of shampoo and conditioner.  I should have ordered them again a few days ago.  I opted to get a mini pc instead of a laptop.  Laptop keyboards, pointer devices and screens are such bad devices that I can't even justify carrying them around with me or spending any money on them.

 

As sad as it was to leave all my possessions behind, including 30 years of digital photos and my life's work, and all but one of my friends, I'm enjoying having a clean start.  Making more is really where the joy is at.  I just hope there is someone out there who appreciates all the devices I left behind.  It would be really sad if that stuff wound up in a landfill way before its time.  I loved all that stuff and only left it behind for a career change.  I can and will build back and it will be arguably better.  In my eyes just different though and mostly a waste.  I can appreciate why my computer from 1999 doesn't work anymore.  The computers I built in the last 10+ years were expected to last much longer than that computer though.

 

I feel a lot of pressure and grief to not buy a musical instrument online.  And then I thought a week ago that I bought my last trombone online from someone in New Orleans and how wonderful that instrument was.  To me, musical recordings are much, much more important than concerts.  These recordings become like friends listening to them over and over.  The melodies, rhythms, harmonies and lyrics sink in over the decades.  I go through phases of intense study of many of them.  I'll often put one recording on and practice playing several different instruments over it for hours.  I don't understand why people enjoy live music so much.  To me, they are just some weird cattle herding event.  My idea about it this last week is that people enjoy getting together with their small group of friends and then being a small cell in the bigger organism of the crowd.  For the musician, it's a place where they can play and then move on so people don't have to hear their same thing over and over for decades.

 

It seems that when you are around a musician, you are getting a performance whether you are paying for it or not.  Musicians spend decades practicing their art without pay.  At least if they are anything like I am.  It's easy to charge for a burrito.  I need that physical thing to survive and if you don't give it to me, then I'll have to fight you for it.  Music is different like that.  Analogies are cool.  Everything is relative.  But as useful as a/b = c/d is, the bridge of the song comes up and takes you to something different.

 

A Farewll to Kings?  For me, I put my trombone on a plane flight, in a nice new case that did not come with that horn, that had a nice red emblem on it that said "King".  And then I did not get on the plane and like everything else I owned, I will not be going back for it because the social situations were so distressing, the stuff wasn't worth it anymore.  I went to the store a few days ago and spent $18 on a Stanley hand saw and $5 on a 1.25" wooden dowel made of pine.  I had a hard time deciding on whether to get the pine one or the hardwood one that was $25.  I figured that I had the saw and could come back if I didn't like it.  My last set was made from an old mop handle and was excellent.  It turns out that I love the new pine set.  I cut it into 3 pieces.  Two are identical and then there is the third which is about 6" longer.  The longer one sounds just above a perfect fifth higher than the other two.  I was waiting for a ride yesterday and there was this wonderful metal rail with several different pieces to it screwed down to a cement wall that was about 2" high.  With my pine sticks hitting them I had the sound of three different pure elements.  It was a wonderful place.

 

I was trying to type on my phone in a car the other day and couldn't do it.  I have been able to play with these big drumsticks in a moving vehicle.  I enjoy using traditional grip in both of my hands playing them on my thighs.  Twirling them comes naturally after a lifetime of twirling cooking utensils in the kitchen.  Playing them while lying in my hotel bed works well too.  Trombone is a silly instrument where the only thing that moves in time is my tongue and my left arm is mostly just locked up holding the thing.  There is this intricate movement of pressure with the left hand/arm though.  Without that pressure control, it would be very difficult to play that instrument.

 

As simple as the trombone seems, it was probably one of the most difficult instruments to make.  The slide has to be absolutely straight.  And how the sleeve on the end of the inner slide works.  It's really a modern marvel.  On the saxophone, you have a flapper with a piece of cork on it.  A spring.  A screw going through a rod.

[Lydia_R]

var rd = 180 / Math.PI;
var dg = Math.PI / 180;

function PiCalculatorDrawLine(ctx, x1, y1, x2, y2, pLineWidth, pColor) {
	ctx.beginPath();
	ctx.moveTo(x1, y1);
	ctx.lineTo(x2, y2);
	ctx.lineWidtdh = pLineWidth;
	ctx.strokeStyle = pColor;
	ctx.stroke();
}
function ShowPiCalculation(pCanvasId, pNumOfSpokes) {
	var canvas = document.getElementById(pCanvasId);
	var ctx = canvas.getContext("2d");
	
	var radius = (canvas.width > canvas.height ? canvas.height / 2 : canvas.width / 2);
	var xCenter = canvas.width / 2;
	var yCenter = canvas.height / 2;
	
	var degrees = 360 / pNumOfSpokes;
	
	// Use Trigonometry SAS (Side Angle Side) to find the third side using the law of cosines (I have the law of cosines memorized)
	
	// c^2 = a^2 + b^2 - 2ab * Cos(C)
	var c = Math.sqrt((Math.pow(radius, 2) + Math.pow(radius, 2)) - ((2 * (radius * radius)) * Math.cos(degrees * dg)));
	
	// the circumference of the circle, estimated by the straight lines on the "c" side of each spoke, is (c * pNumOfSpokes), which whould approximately equal (be a little less than) 2 pi r
	
	var approximatedCircumference = c * pNumOfSpokes;
	var pi = ((approximatedCircumference / 2) / radius);
	
	alert(pi);
	
	// OK, that worked!  First time :-)
	// Now draw it!
	canvas.style.display = "block";
	
	for(var a = 0; a < pNumOfSpokes; a++) {
		var color = "red";
		if(a % 2 == 0) color = "green";
		
		var baseDegrees = a * degrees;
		// x1 and y1 are xCenter and yCenter
		var x2 = xCenter + (Math.cos(baseDegrees * dg) * radius);
		var y2 = yCenter - (Math.sin(baseDegrees * dg) * radius);
		PiCalculatorDrawLine(ctx, xCenter, yCenter, x2, y2, 1, color);
		
		// now draw the other side of the wedge (assumming standard position (to the right) and going up
		var x3 = xCenter + (Math.cos((baseDegrees + degrees) * dg) * radius);
		var y3 = yCenter - (Math.sin((baseDegrees + degrees) * dg) * radius);
		PiCalculatorDrawLine(ctx, xCenter, yCenter, x3, y3, 1, color);
		
		// now connect the outer two points
		PiCalculatorDrawLine(ctx, x2, y2, x3, y3, 1, "blue");
		
	}
	
}

 

/* Lydia R 2024-10-17 Calculation of pi Through Methods of Exhaustion Javascript Demo 2024-10-18 Oh My! I used pi in my calculation of pi!! LOL! How to compute it otherwise? with a right triangle, we would know all three angles. We do know all three angles because the other two of the non-right triangle is (180 - AngleC) / 2. How do angles help us without trig? You can express them in rise over run. I'll work on this some more. This was the conclusion I ran into months ago and why I didn't code it then. It's certainly a linear algebra thing and the code in my collisionDetection.js script may help. */

 

 

A few days later here and I convinced again that I need some "real world" measurement to continue with this equation. (Remember, this is one of my hobbies, looking for this answer on my own.) I could draw a theoretical line between the two lines in the spoke to split it into two right triangles. I would know the degrees of all three angles and it would be a right triangle, but I still only have the measure of that one site. To use trig, I would be using pi and the trigonometric function.

 

My current working theory is based on knowledge of a 3-4-5 triangle. If I call the radius the 5 side, I can compute the rest with ratios. 20 years ago I studied that and drew it out and counted the squares. But in this case, I'm stuck with the angle being 36 degrees (shown in these screenshots I just took). And of course with that 3-4-5 stuff, it's like a lookup table as well with those known constants.

 

I learned trigonometry without a calculator. I'm still having fun working this problem.

 

 

[Lydia_R]

I've been crunching numbers and forming theories during the election this fall.  I did some rough calculations of pi with both a circumference measure of a clock and counting the squares in the graphing program I created.  Both seem to point to pi being 3.1418.

 

My political theories, if you want to call them that, have become intense.  Being born in 1971, I grew up in the space age.  And I've lived most of my life in cities where it is fairly hard to see the stars.  I did see a meteor shower 30 miles north of Memphis TN in 1995, but it was right next to the Navy base that I was working at and I've come to realize that it could have been faked.  It could have been a spacecraft falling apart.  We saw what looked like a rock falling too.  It was less than a mile over our heads.  Still, I saw a jet fighter a couple months ago fly straight up into the blue sky until I couldn't see it.  I've seen what looked like planets in a plane before.  I don't know how hard that would be to fake.

 

Theories that the Chinese are far ahead of us in their "closed" community.  If they hid their technology, they could have had computers and such hundreds of years ago.  They could have bombed the moon to cause all those craters.  If they didn't and the planets really do exist like I have been taught all my life, then it is likely inevitable that Mercury will hit the sun someday.  If it does, my theory is that it will throw off projectiles in the plane of the solar system and that is what causes the craters on the moons and planets.  

 

Wireless communication has been around way before I was born.  TV, radio.  My dad worked in the industry before I was born.  I question how easy it is to communicate through space.  I've been thinking about optical telescopes on Earth pointed at computer monitors on the moon as a way to relay information.  I read on Wikipedia that there was a transit of Mercury across the sun in 1769 that was used with trigonometry to estimate the distance of the sun.  I haven't put thought into the math of how that would work, but it is interesting that the date of that was just a few years before 1776.  Without an event like that, we'd likely need to triangulate from the moon to make any precise measurement.  I mean, I'm sure we could setup a couple points on Earth and get extremely accurate with the angles and measurement between them, but we'd really need a precise calculation of pi to be accurate.  Is it possible?  With what degree of accuracy?

 

I've become ridiculously untrusting of science talk and publications lately.  At worst, this is a fun hobby of mine.

 

My latest theory is that the majority of the carbon we burn and that which comes from the Earth naturally, goes up to the moon.  Once it is mostly up there, it makes sense that global warming would take over and the water would go there next.  The idea that the moon would start spinning and the Earth would slow down.  That they would Trade Places.  That it is a cycle like that.

 

How would you know how far a star is away?  By the speed of light coming from it?  In comparison to other stars?  In comparison to the sun or our moon?  From the big circle of the Earth on one side of the sun and the Earth on the other side of the sun?  It reminds me of the Apollo 13 movie.  "All we need is a fixed point in space, right?"