trigonometr

This is my trick for remembering trigonometric values. Forget memorizing those little charts or triangles, this is so much easier. 

I learnt this in 10th grade from my favorite teacher (who wasn’t actually my maths teacher at the time) and I still used it for the rest of High School and will probably continue to use it in University.

So what do you do? You draw the chart in the picture. That one. 

OK, but how do you read it? Like a table. (If you’re unfamiliar with radians, do not worry. It also works in degrees, just write: 0, 30, 45, 60, 90 instead.)

For example, if you want to find the sin(pi/2), you simply start at sin(x) and then move over until you’re at pi/2 (or 90 degrees). It says sqrt(4). Now you put that over 2, and you have it. Sin(pi/2) = (sqrt(4))/2, also known as 2/2, or 1. 

If you want to find the cos(pi/4), you do the same thing: start at cos(x), move until (pi/4). It says sqrt(2), now put that over 2, and you have cos(pi/4) = sqrt(2)/2.

For tan(x), just take the sin(x) value and divide it by the cos(x) value!

So tan(pi/6) = sqrt(1) / sqrt(3) = 1/sqrt(3).

I find this so easy to remember because the first line starts at zero and increases, and the second line starts at 4 and decreases. It works for any values (yes, even ones above pi/2!) and it’s so simple. I encourage you to start by writing it out on every paper, when you’re studying, and on your exams. 

Drawing process for the sine function

Since people were really interested on the sine function in that previous image, I figured I’d post this one as well. It will explain things better.

This is what the sine function of the circle actually means: it’s the y-coordinate associated with the arc length, shown in blue. A circle has circumference 2·pi·r. The unit circle has radius 1, so the full unit circle has circumference (or arclength) 2·pi.

The blue arc and the blue line shown at right have exactly the same length. This is the angle in radians.

This is the geometric definition, though. A much more powerful definition is based on infinite series.

(For the polygonal trig functions, I kept the idea of an angle around the polygon, but I gave up on the idea of using the length around the polygon - its perimeter. This caused a lot of confusion about why the side of the square didn’t trace a straight line.)

The sine and cosine functions for the circle, as every student should see them.

(Edit: the animation is also available, without watermark, at higher resolution and slower frame rate at Wikimedia Commons.)

HAPPY PI DAY! To celebrate, here’s this long-due animation of the usual trigonometric functions, sine and cosine, geometrically defined in terms of the unit circle.

I know this animation is a bit of   the same   as several   others   of my previous   animations  , but this is THE version that I should have done ages ago, if not done first of all.

This is what the sine and cosine functions, the ones you are taught, really are in terms of the unit circle.

First, we have the unit circle (with radius = 1) in green, placed at the origin at the bottom right.

In the middle of this circle, in yellow, is represented the angle theta (θ), that we’re going to plug in our trigonometric functions. This angle is the amount of counter-clockwise rotation around the circle starting from the right, on the x-axis, as you can see. An exact copy of this little angle is shown at the top right, visually helping us define what θ is.

At this angle, and starting at the origin, we trace a (faint) green line outwards. This line intersects the unit circle at a single point, which is the green point you see spinning around at a constant rate as the angle θ changes, also at a constant rate.

Now, we take the vertical position of this point and project it straight (along the faint red line) onto the graph on the left of the circle. This gets us the red point. The y-coordinate of this red point (the same as the y-coordinate of the green point) is the value of the sine function evaluated at the angle θ, that is:

   y coordinate of green point = sin θ

As the angle θ changes, we can see the red point moves up and down, tracing the red graph. This is the graph for the sine function. The faint vertical lines you see passing to the left are marking every quadrant along the circle, that is, at every angle of 90° or π/2 radians. Notice how the sine curve goes from 1, to zero, to -1, then back to zero, at exactly these lines. This is reflecting the fact sin(0) = 0, sin(π/2) =1, sin(π) = 0 and sin(3π/ 2) = -1

Now, we do a similar thing with the x-coordinate of the green point. However, since the x-coordinate is tilted from the usual way we plot graphs (where y = f(x), with y vertical and x horizontal), we have to “untilt” it in order to repeat the process above in the same orientation. This was represented by that “bend” you see on the top right.

So, the green point is projected upwards (along the faint blue line) and this “bent” projection ends up in the top graph’s rightmost edge, at the blue point. The y-coordinate of this blue point (which, as you can see due to our “bend”, is the same as the x-coordinate of the green point) is the value of the cosine function evaluated at the angle θ, that is:

   x coordinate of green point = cos θ

The blue curve traced by this point, as it moves up and down with changing θ, is the the graph of the cosine function. Notice again how it behaves at it crosses every quadrant, reflecting the fact cos(0) = 1, cos(π/2) = 0, cos(π) = -1 and cos(3π/2) = 0.

And there you go. That’s all there is to it. That’s what sine and cosine are. Simple, huh?

Now, while the concept itself is pretty simple, a lot of people get confused about what the sine and cosine functions actually represent, because visualizations such as this are not presented to them when they are first taught trigonometry.

A lot of teachers, and plenty of school books, fail to mention any of this in detail, as I tried to do here, instead throwing a bunch of formulas in front of students. But the geometric intuition, as presented here, is much simpler to grasp, much more useful in general, and will stick to you for life once you get it. The formulas and important values for sine and cosine don’t need to be memorized anymore, because now you should understand what these values should be, given the underlying logic of things. And that’s what math is all about: making sense of things so they are plainly evident to anyone.

In my most popular post to date (over 360 thousand notes as of now, holy crap!), I saw a lot of people commenting that seeing the top graph, which is the sine function for the circle, made all that trigonometry stuff click.

I was baffled. People were angry that no teacher has ever showed anything like that to them before. That’s crazy! At this age where computers are everywhere, this sort of thing should be in every classroom, and be seen by every student.

So, in order to do justice to the unit circle and these immensely important trigonometric functions, and in order to fill an obvious pedagogical hole in math classrooms and textbooks everywhere, I decided to finally make this animation. No fancy or crazy alternative takes on the sine and cosine this time, just the good ol’ pair of trigonometric functions we all should understand and love.

Happy Pi Day, everyone!

2

The Magic Hexagon for Trig Identities

This little trick saved my life last year. Basically, you draw the hexagon and it will help you remember the trig identities. 

  • When you go around it, you get the quotient identities 
  • When you go across the 1, you get the reciprocals 
  • When you go clockwise around three of the triangles, you get the pythagorean identities

Read the rest here, there is a lot more!!!

My life is measured by the due dates of projects and homework squeezed into the margins of a worn down agenda. It’s measured by the weight gain that accompanies the two weeks leading up to midterms and finals. It’s measured by the nights I lie in bed at 2 am, trying to remember whether the day is in fact, truly over yet.
I am measured by numbers. It’s something I’ve never let myself forget.
There’s a two by four rectangle in the corner of my room that houses me, nine textbooks, and five binders. We’re competing for space, and somedays I feel as though they are winning. Post-it notes on the bathroom mirror with the anti-derivatives of trigonometric functions and the rules for naming complex ions are a more familiar sight to me than my own father’s face.
My mother and I don’t interact with each other, except for our twice-daily screaming matches that make our house shake on its fifty-year old foundation. I don’t mind that, I guess. It’s hard to make conversation with a stranger who refuses to learn anything about you except your GPA.
My father and I didn’t speak for four days after my SAT score came out and for those four days, there was a part of me that missed the man who had been my best friend since I was three and a half, but an even bigger part that was relieved that I didn’t have to hear his disappointment in words the way I could see it on his face.
I’m sixteen and I’ve never kissed a boy the way they do in the movies, with tongue and passion, which really doesn’t feel like that much of a loss until I remember that my cousin met her husband when she was 15 and great-grandmother probably had two children with a third one on the way when she was my age. It was a different world, I tell myself.
I do not know what fuels me. My only ambition is to reconcile with the restless soul inside me and some days even that seems like a hopeless cause. Maybe I am not destined for happiness. Some women just aren’t.
But I also know that I am not destined to be this. I am not destined to be a doll, a faceless doll, with a barcode on my back.
I know that numbers do not define me.
My current grade in chemistry is not my intelligence. My weighing machine can not give you an accurate estimate of my beauty. The number of extracurriculars I am involved in will not tell you whether or not I am interesting person. My age is not synonymous with “naive.”
I am a person, not a collection of numbers.
Do not try to simplify me into a binary code. Maybe you are content leading a life with 0s and 1s but I live in a world with words and colors and sounds and the feel of sunshine on my face.
I am sick of people trying to take these things away from me and throwing poor substitutes back at me. I am sick of numbers that attempt to sort me and categorize me and place me where I supposedly belong. I am sick of people telling me that I need to live in the real world because it is you who do not understand what the real world is.
I have news for you. The real world is here and now, and it is full of people with hopes and dreams and ambitions and thoughts that you do not know about because you spend too much time trying to box them into your neat little columns.
Maybe I do not want to fold gracefully into the space you have left for me in those cramped and crowded boxes. Maybe I want to spread out and take over the entire page, leaving spaces and gaps in whatever quality I don’t deem worthy to define me. Maybe I do not want to be as simple as you try to make me.
Maybe I am not a number.
Have you ever considered that?
—  I Am Not A Number, the-ink-stained-heart

me: fuck this whole societal system. I was born into a preconstructed cage. we all were. I belong in the place where we have no physical limitations and no dream limitations. everything is a fucking limitation here. we’re all trapped. we’re all stuck in the cave, knowing nothing else but the shadows dancing on the walls. we think it’s fine but is it really? somewhere out of all the galaxies we could be in we’re in a place where people kill each other and steal and lie and terrorise and hurt even after we have mini computers in our hands? what is this? why can’t we fly or swim with dolphins every day or run like the wind? why do we have to learn the trigonometric equations only to forget them all after a standardised test? why are people prized by high achievement but not the ones who work the hardest? why are we forced to go to school and get a boring job and then retire and die? why can’t we do our hobbies and get good at that and then make a living out of that instead of wasting time on things we don’t enjoy? what’s with all the stigma with mental illnesses and suicide and even wearing what we want? why are we are trapped? this society is a mess. and we’re all doomed to these walls and there’s no way out because everyone is fine with it??

Also me, 5 mins later: LOL what!??? I was just kidding lmao, everything is fine, I’m ok!! don’t mind me :))
The History of Trigonometry

From LiveScience.com:

Trigonometry follows a similar path as algebra: it was developed in the ancient Middle East and through trade and immigration moved to Greece, India, medieval Arabia and finally Europe (where consequently, colonialism made it the version most people are taught today). The timeline of trigonometric discovery is complicated by the fact that India and Arabia continued to excel in the study for centuries after the passing of knowledge across cultural borders. For example, Madhava’s 1400 discovery of the infinite series of sine was unknown to Europe up through Isaac Newton’s independent discovery in 1670. Due to these complications, we’ll focus exclusively on the discovery and passage of sine, cosine, and tangent.

Beginning in the Middle East, seventh-century B.C. scholars of Neo-Babylonia determined a technique for computing the rise times of fixed stars on the zodiac. It takes approximately 10 days for a different fixed star to rise just before dawn, and there are three fixed stars in each of the 12 zodiacal signs; 10 × 12 × 3 = 360. The number 360 is close enough to the 365.24 days in a year but far more convenient to work with. Nearly identical divisions are found in the texts of other ancient civilizations, such as Egypt and the Indus Valley. According to Uta Merzbach in “A History of Mathematics” (Wiley, 2011), the adaptation of this Babylonian technique by Greek scholar Hypsicles of Alexandria around 150 B.C. was likely the inspiration for Hipparchus of Nicea (190 to 120 B.C.) to begin the trend of cutting the circle into 360 degrees. Using geometry, Hipparchus determined trigonometric values (for a function no longer used) for increments of 7.5 degrees (a 48th of a circle). Ptolemy of Alexandria (A.D. 90 to 168), in his A.D. 148 “Almagest”, furthered the work of Hipparchus by determining trigonometric values for increments of 0.5 degrees (a 720th of a circle) from 0 to 180 degrees.

The oldest record of the sine function comes from fifth-century India in the work of Aryabhata (476 to 550). Verse 1.12 of the “Aryabhatiya” (499), instead of representing angles in degrees, contains a list of sequential differences of sines of twenty-fourths of a right angle (increments of 3.75 degrees). This was the launching point for much of trigonometry for centuries to come.

The next group of great scholars to inherit trigonometry were from the Golden Age of Islam. Al-Ma'mun (813 to 833), the seventh caliph of the Abbasid Caliphate and creator of the House of Wisdom in Baghdad, sponsored the translation of Ptolemy’s “Almagest” and Aryabhata’s “Aryabhatiya” into Arabic. Soon after, Al-Khwārizmī (780 to 850) produced accurate sine and cosine tables in “Zīj al-Sindhind” (820). It is through this work that that knowledge of trigonometry first came to Europe. According to Gerald Toomer in the “Dictionary of Scientific Biography 7,” while the original Arabic version has been lost, it was edited around 1000 by al-Majriti of Al-Andalus (modern Spain), who likely added tables of tangents before Adelard of Bath (in South England) translated it into Latin in 1126.

Read More

For people looking for History of Mathematics resources!

link submitted by grnephrite

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Studdiction’s weekly roundup - 16th Aug

My posts: 

Favourite posts and resources of the week:

youtube

View the TED-Ed Lesson How do we measure distances in space? - Yuan-Sen Ting

When we look at the sky, we have a flat, two-dimensional view. So how do astronomers figure the distances of stars and galaxies from Earth? Yuan-Sen Ting shows us how trigonometric parallaxes, standard candles and more help us determine the distance of objects several billion light years away from Earth.

ok so in pre-calculus we’re working on trigonometric functions (sin, cos, tan, etc) and today we had work time 

so naturally i went over to a friends desk and we started discussing how much of a difference her boyfriend’s new haircut made on his overall vibe

and the teacher overheard and asked me in particular what hair had to do with pre-calc

so i paused for a moment and said “take for example, sir, my hair.”

“does it not resemble the function sin(x)?”

and he just gave me a vacant look but i could just feel that i’d won

so since I’m probably getting pet rats in a couple weeks and I’m excited for them, what if all the Pines had their own very close pet, not just Mabel with Waddles?

Dipper would, I think, do a ton of research before selecting a strange, somewhat exotic pet that fits his personality. Maybe an African Grey parrot, known to be super smart, that Dipper spends hours trying to teach intelligence puzzles and increasingly complicated phrases? Imagine Dipper using his parrot as a study aide, pretending its a person he’s teaching the content to (they say teaching something is the best way to properly learn it) and this leads to amusing situations where the parrot starts including trigonometric functions and the steps of mitosis into its everyday randomly repeated vocabulary

Stan would probably be much more “boring” by comparison, having taken pity on some stray cat or dog (grudgingly impressed by its moxie in breaking into the shack or something). The stray is missing half an ear and is clearly blind in one eye with a crooked tail, looks ugly as sin and hates everything and everyone except for Stan. They learn to tolerate family, but they will never cuddle and ask for attention from the other Pines like Waddles sometimes does. 

This is totally coloured by my recent experience, but I like the idea of Ford having a rat. Normally, rats are always better in pairs, but Ford just has the one. It makes up for this species isolation by literally always being with Ford, somewhere in his jacket or sleeve usually. Rats also only live at most to 3 years, but Ford had it out of the portal, and its still pretty spry after a couple years on the Stan o War. Stan’s convinced it’s some genetically altered specimen Ford rescued, because there’s no way Ford has a normal pet rat. Dipper thinks its a normal rat and that Ford is doing Great Science Things to prolong its life. Mabel thinks that it’s “love conquering all” and that karma really owes it to Ford anyways. The real truth is that it’s a pet rat that happens to be from a dimension where rats have a much longer lifespan and are even more intelligent