Masthash

# #trigonometry Method vs trick according to George Polya

ChatGPT agrees that my half-angle approach is a method rather than a trick.  Earlier I had said that the half-angle formulas are an essential ingredient in the derivation of Viète's $$\pi$$ formula. According to Eli Maor, Viète's formula marks the beginning of mathematical analysis, and Jonathan Borwein calls its appearance "the dawn of modern mathematics".

The work described in the link below is another example of the centrality and versatility of the half-angle formulas (although the authors mistakenly call them double-angle formulas), a centrality that transcends the field of geometry and trigonometry in the plane.

https://www.sciencedirect.com/science/article/pii/S0021904513001159?via%3Dihub#br000045 #Jupiter in #3D with #CSS #trigonometry https://codepen.io/thebabydino/pen/GRYbVoe
Rotates around y axis, scroll changes x axis tilt if browser supports CSS #scroll animations.

Just one image used for sphere surface (#2). Distortion from rectangle to "dress" sphere is all CSS. If you're a patron on Patreon, then you may have seen the concept in action last month, though not with Jupiter.

Also: very impressed with #Firefox - 3D perf used to be horrendous. Now it's better than #Chrome!
#CodePen #maths #geometry   Just released: Basic Trigonometry Cheat Sheet by Bunno

Here's their description of it: My personal notes for some basic trigonometry last minute before a test, feel free to use it.  An interesting and original (I think) puzzle from Micky Bullock.

"Last week Negligent Neil calculated length AC. He had forgotten to switch his calculator from radians to degrees but, fortunately, he still got the answer right.

"An inky splodge has now obscured the angle at A. Negligent Neil has forgotten what his answer was for length AC, but he insists it was between 40 cm and 50 cm.

"Find length AC to 3 significant figures."  In addition to #algebra, al-Khwarizmi made significant contributions to the fields of #astronomy & #trigonometry

Al-Khwarizmi's astronomical observations / calculations helped refine the solar calendar & contributed to development of accurate timekeeping devices.

He compiled detailed astronomical tables, providing information on the movements of the #sun, #moon & #planets.

These tables were widely used by astronomers in the Islamic world.    Touched by an Angle

A show about people learning #trigonometry when an guardian #angle intervenes into their lives.  How did I miss this news? Particularly considering that I follow Big Math Nerd @MerlinJStar ? Two Black Girls proved the #PythagoreanTheorum using #trigonometry
https://www.scientificamerican.com/article/2-high-school-students-prove-pythagorean-theorem-heres-what-that-means/ Wondering how many places in the world teach the unit circle in trigonometry class (at the college level). Just pushed another video to my Trigonometry Playlist on YouTube.

It was a shorter one this time bit still an important one!

Half-Angle Identities 👉

https://youtu.be/sRruOq_tomQ Apollonius had a great name and an even better theorem.

Check out my newest math vid on how to find the length of a triangles median 📐

https://youtu.be/6yGHW7Qj4Vc Curious how you can find the length of a triangle's median?

Appollonius has you covered ✅️

https://youtu.be/6yGHW7Qj4Vc You need to remember 3 numbers and you can piece together the entire Unit Circle.

I saw the 3-line visualization somewhere a while ago and wanted to recreate it. Here it is!

https://youtu.be/Y1iEwAc6R_c I can now have stuff like this in #CSS!

If you're wondering what in Satan's name is that supposed to be doing... well, creating a sphere like in the second image.

Does it work? Not with that formula, as it would need division with units in calc() to get an angle from a triangle where I know the hypotenuse & opposing edge.

But smth like asin(3vw/5vw) doesn't work!

So I have to use asin(3/5), then multiply with 1vw to set lengths.   NEW PROOF FOR SIN (X+Y)

Several interesting proofs of this identity are given on MathSE. In this note, I will provide another proof that is possible because, although traditionally presented in textbooks as a consequence of the angle sum identity sin(x+y), the double-angle formula for the sine, sin(2x)=2sin(x)cos(x), can be derived independently of it. Then, as Blue has pointed out, the fact that supplementary angles have the same sine is an easy consequence of the double-angle formula. My wake-up call for the extraordinary explanatory power of half-angle formulas has reached 3,000 views on MathSE. Are the half angle formulas more fundamental than the Pythagorean theorem or the law of cosines?
#math #trigonometry #geometry #halfanglesmatter #symmetrymatters    Yo defendiendo que las fórmulas de medio ángulo son más fructíferas que la ley de cosenos.  Made a Lissajous curve generator in #pico8 .
Draw curves by setting the frequency of the vertical and horizontal oscillator.  Stuff I've been doing lately: support* tests for mathematical functions.

Confused about what browser supports what mathematical functions or constants?

https://codepen.io/thebabydino/pen/yLRBJXP tells you what your browser supports.

*not only support, but more about that/ use cases another time    Awesomeness of the day!

The #Pythagoreantheorem has been proven before in other mathematical disciplines such as algebra and geometry, but never through #trigonometry.
Most mathematicians believed it was not possible.

Two truly amazing #HighSchool students from NOLA may have just changed that!

Many thanks @randulo and @lorriberri
for sharing! ♥️ Lemniscate of Bernoulli has the equations:

x = (a·cos(t))/(1 + sin²(t))
y = (a·sin(t)·cos(t))/(1 + sin²(t))

where a is the lemniscate half width and 0 ≤ t ≤ 2·π 😊

A couple of years back, I made a canvas lemniscate of Gerono animation 😼 https://codepen.io/thebabydino/pen/yLyxgVQ

Feel free to guess as to why I've posted this stuff 🤭😈  #Development #Techniques
Improving CSS Shapes with trigonometric functions · We now have a lot of ways to work with CSS Shapes https://ilo.im/1239qq · by @DanCWilson The other day I looked up at the half #moon in a blue sky and saw a pair of bald eagles circling between it and me. They were about 1/3 its diameter, white heads and tails distinct even at that distance.

The diameter of the moon as seen from earth is 0.5°. The wingspan of a #BaldEagle is about 2m. (They’re huge!) These birds were about 2/tan(0.17) = 688 meters away from me. Based on the elevation angle, they were soaring about a third of a mile high.

A magical moment. I keep seeing news articles claiming some high school students invented a #trigonometry proof of the Pythagorean Theorem. But for some reason, their paper isn't available online at the place it should be: https://meetings.ams.org/math/spring2023se/meetingapp.cgi/Paper/23621

I wonder also if they referenced this Jason Zimba proof from 2009: https://forumgeom.fau.edu/FG2009volume9/FG200925.pdf

Has anyone in the #math world have a copy of the paper? Ramon Antonio Vargas: US teens say they have new proof for 2,000-year-old mathematical theorem: New Orleans students Calcea Johnson and Ne’Kiya Jackson recently presented their findings on the Pythagorean theorem US teens say they have new proof for 2,000-year-old mathematical theorem

New Orleans students Calcea Johnson and Ne’Kiya Jackson recently presented their findings on the Pythagorean theorem

https://www.theguardian.com/us-news/2023/mar/24/new-orleans-pythagoras-theorem-trigonometry-prove Two black, female high school students in Louisiana have just proved Pythagorus' theorem using trigonometry, which was thought to be impossible. #RonDeSantis will now ban #Pythagorus from the curriculum in #Florida on the grounds that he was too woke.

https://www.theguardian.com/us-news/2023/mar/24/new-orleans-pythagoras-theorem-trigonometry-prove US teens say they have new proof for 2,000-year-old old mathematical theorem | New Orleans | The Guardian
https://www.theguardian.com/us-news/2023/mar/24/new-orleans-pythagoras-theorem-trigonometry-prove Just released: Trigonometry Questions Year 9 Cheat Sheet by inkirbythesecond

Here's their description of it: trigonometry questions + answers for year 9 students  But what excites me the most about these formulas is their potential to revolutionize the way we teach trigonometry to high school students. By introducing the half-angle formulas, we can help students better understand the development of the formulas that derive from them. This is where the half-angle formulas truly shine and can make a significant impact on math education.

GeoDom: The Half-Angle Formulas: A Powerful Tool for Trigonometry Students http://geometriadominicana.blogspot.com/2023/03/the-half-angle-formulas-powerful-tool.html?m=1 But now, we can compute the scaling factor as the cosine of the skew angle, animate the skew angle* and the scaling factor changes accordingly 🥳🎉

*@​property support needed  So we must scale down edges along the y axis post-skew in order to get a rhombus again!

This scaling factor is the cosine of the skew angle, which doesn't change linearly with a linear change in the angle from -90° to 90°. We can approximate the change with cubic-bezier, but we can't use individual transform properties as those would place the skew after the scale.

So when I first coded this 5 years ago, I chose to use #JS.   There's also this square tiling https://codepen.io/thebabydino/pen/JjBEzMQ

This is a pure #CSS version of something I coded with #JS a few years back https://codepen.io/thebabydino/pen/JNZGwE/

If you look at the orange squares, they get squished to rhombi & eventually to a line (then they go back).

This is done with skew along the x axis. But the way this works https://codepen.io/thebabydino/full/JoQRdb means edges along the other axis (y) get longer & we don't have a rhombus (all 4 edges equal) anymore!   Also made this torus knot https://codepen.io/thebabydino/pen/vYdMVJK??editors=0100

CSS trig functions allow to avoid hardcoding the position vector into a custom prop for each & every item making up the knot as I had to before when coding such shapes:  GRUB is giving me a massive headache today😵😵‍💫, so here are a few things I made on #CodePen
that should now work in Chromium 111+ as a result of support for trigonometric functions dropping.

First up: Animated Möbius strip using CSS mathematical functions https://codepen.io/thebabydino/pen/wvybyMo  #Development #Introductions
Trigonometric functions in CSS · Calculate the sine, cosine, tangent, and more in CSS https://ilo.im/11k7dw So a YouTube video got me thinking about numerical methods for calculating sines and cosines. Not that it's important for most purposes, because essentially every programming language and math-capable utility has built-in sine and cosine functions.

Nevertheless, if you wanted to write the code yourself, you would at some point be doing a Taylor Expansion for sine and cosine according to the usual formulas. The problem is getting the expansions to converge to accurate, reliable values in as few terms as possible. Here is what I would recommend:

1) Angles greater than pi or less than -pi, add or subtract multiples of 2*pi to bring them between -pi and pi.

2) Use symmetry to map the angle to the range 0 to pi/2.

3) Angles from pi/4 to pi/2, you can calculate as the complementary function on the complementary angle. In other words, if you want to figure out the sine of 85 degrees, you can instead calculate the cosine of 5 degrees. So that reduces all our calculations to angles from 0 to pi/4.

All of that is obvious. Here's the part that is less obvious:

4) Figure out the sine and cosine of angles 0, pi/16, pi/8, 3*pi/16, and pi/4. As in, pre-calculate them, and store them as constants in your program. Now remember these trig identities:

sin(a+b) = sina*cosb + cosa*sinb

cos(a+b) = cosa*cosb - sina*sinb

Those identities will let you calculate all your angles as offsets from 0, pi/16, pi/8, 3*pi/16, or pi/4, whichever is closest. Like, suppose you wanted to calculate sin(7*pi/64). Well, think of that as sin(pi/8 - pi/64). Based on the trig identities above, that'd be sin(pi/8)*cos(pi/64) - cos(pi/8)*sin(pi/64). And remember, we have the sine and cosine of pi/8 pre-calculated, so all that's left is doing the Taylor Expansions on cos(pi/64) and sin(pi/64), which will resolve in only a few terms.

You could take this thinking further, where you're doing offsets against multiples of pi/32 or even pi/64. Whatever lets you get an accurate result in only a handful of expansion terms Also made a scroll-driven angled slide for the same #CodePenChallenge - see it on @CodePen https://codepen.io/thebabydino/full/zYJKRyq

Uses pure #CSS scroll-driven animations to make angled content section also slide right on vertical scroll.

Uses CSS shapes to make text flow along angled edges.  Made a pure #CSS thing for the angled #CodePenChallenge https://codepen.io/thebabydino/pen/ExePBZp

✨tan to compute vertical spacing
✨responsive, no angle change on resize
✨no section background distortion (as skew does)
✨heavily commented
✨interactive angle change if browser supports tan()  #OnThisDay Birth Anniversary of Italian #Physicist Alessandro Volta (1745) - credited as the inventor of the electric #Battery.

Birth Anniversary of Persian polymath Nasir al-Din al-Tusi (1201) - considered the creator of #Trigonometry as a mathematical discipline.

Today is World #PangolinDay and #PlutoDay.

https://knowledgezone.co.in/news   RT ics-kitagawa
CSS trigonometry demo 2/3
See demo on codepen↓
#css #trigonometry
https://codepen.io/ics-kitagawa/pen/eYjbwMq  #trigonometry

cos 9° = √2/8 + √10/8 + √(5−√5)/4
cos 15° = √2/4 + √6/4
cos 21° = √2/16 + √6/16 + √10/16 + √30/16 − √(5 − √5)/8 + √(15 − 3√5)/8
cos 45° = √2/2
cos 57° = −√2/16 − √6/16 + √10/16 + √30/16 − √(5 + √5)/8 + √(15 + 3√5)/8
cos 69° = √2/16 − √6/16 + √10/16 − √30/16 + √(5 − √5)/8 + √(15 − 3√5)/8
cos 75° = −√2/4 + √6/4
cos 93° = √2/16 + √6/16 − √10/16 − √30/16 − √(5 + √5)/8 + √(15 + 3√5)/8

cos 93° − cos 57 =
cos 9° + cos 21° − cos 69° =
cos 15° + cos 75° = I made a pure #CSS thing on #CodePen 🖤 https://codepen.io/thebabydino/pen/gOjGyby

Regular dodecahedron (12 ⬠) expands to rhombicosidodecahedron (20 △ in place of former vertices, 30 ◻ in place of former edges), twists to snub dodecahedron (80 △ & 12 ⬠), twists back, collapses to icosahedron (20 △).

Rhombicosidodecahedron https://mathworld.wolfram.com/SmallRhombicosidodecahedron.html

Snub dodecahedron https://mathworld.wolfram.com/SnubDodecahedron.html  When al-Kkwarizmi's work was translated from #Arabic into #Latin, the translators mistook جيب, a maths-specific term, for جيب beaning 'bosom', and translated it as the Latin word for 'bosom', which is 'sinus'. That's where we get the term 'sine' from.

This means that whenever you've done #trigonometry, you've been talking about breasts without even realising it.

/end Many ancient mathematicians wanted to figure out how to calculate the length of a chord, or a line joining two points on the circumference of a #circle. In #India, this was accomplished by drawing a circle of radius 3438 units and measuring the chords which subtend various angles.

The diagrams reminded mathematicians of a bow, and so this measure was called jya (ज्या), the Sanskrit word for 'bow'.  I’ve got my issues w/ this platform, but the ppl who choose not to be overly technical (or any of the other horrible things) are very nice & helpful.

I wish I’d had this for middle school math. Somebody in the #Fediverse would’ve been able & willing to help me with #trigonometry. I remember thinking a life of crime would be a better choice because they give you 3 meals a day in prison. (Like many kids, I was uninformed about the realities of our incarceration system 😕)   Came across https://austinhenley.com/blog/cosine.html

... and remembered that 10 years ago, I also did something similar in Sass https://codepen.io/thebabydino/pen/DEvpOP for inverse trigonometric functions (I had trigonometric functions using Compass, but not even Compass had the inverse ones), including the range reduction on a subsequent pass https://codepen.io/thebabydino/pen/VwPBwo to reduce errors.

Fast forward to today, no need for any of it anymore, there's sass:math! https://sass-lang.com/documentation/modules/math  @codewiz
I have a feeling #math teachers could look at the #demoscene in general and #glsl #shaders in particular to help make #trigonometry, matrices and volumetric functions more appealing through tools such as #ShaderToy.