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はてなキーワード: sinとは
（１） sinθ cosθの 定義を述べろ。 （２） sin(α＋β） cos(α＋β）を証明せよ、 というものだった。
教科書の定義を述べさせて、 定理を証明させるもので、 純数学的な問題で、教育委員会に激震が走った。
定義と言えば、 ベクトルの内積の定義すら分かっていなかった学生が多い時代だったので、ショックだった。
（１）は、 単位円の ｘ 座標が sinθで ｙ座標が cosθである。 このように定義する。 すると、三平方の定理により、 sin^2+cos^2=1 は自明である
一般に高等裁判所は、 刑事訴訟法３９６条により控訴を棄却するが、 訴訟費用の不負担は、 刑訴法１８１条１項但書を適用して、行う。 ここで、 「により」 と、
高等裁判所の裁判官は、 福岡高裁では、令和４年４月の、 岩坪朗彦が有名だが、 岩坪朗彦は、 民事訴訟法の３つの条文により棄却した。ここで、
この論文は最初は、「ある種の不定方程式の解に関する研究」として開始されたが
x^p+y^p=z^p に対しては、クンマーという教授が、６７％ではＯＫであるというイデアル理論という珍奇理論によりやったが、全部証明ではないということで、解釈が間違っているから
両辺をｚで割って、
楕円関数と有理数点で扱って、 有理数体と、楕円関数の複素関数上のモジュラー性を考えないといけない。
予想（Conjecture）であってまだ定理かどうか分からなかったということです。 １９５０年代の数学者は、定理には、驚愕性が必要だが、このコンジェクチャーにはそれが備わっていないということで
それはいいとしても、高等学校程度で習う定理だと、 sinの加法定理とかがありますが、 sin(α＋β）は書き下せるというものである。 sinとcosには、 sinθ^2＋cosθ^2＝１
という有名な関係がある。
何で東京大学で、 sinx/x がそのうち出るだろうと予備校の先生の間で予見されたかというと、平成１１年に、教科書に書いてあってあえて証明がついていない、
sin(α＋β） ＝
cos(α＋β） ＝
ちょっと右の方は割愛しますが、展開公式があってこれを証明しなさいという問題が出たので、 sinθ/θは、 証明するか、結論を述べるかの問題が出るだろうと思っていたときに
平成３０年に、実質的には出たということです。 sinθ/θは、 教科書に書いてある有名な定理で、証明も一応、学習参考書についているはずなので、そのうち取り上げられるだろうと
予想されていたが遂に出たのかということになったわけであります。
東京大学は官僚養成学校なので創作応用問題が多く、いわゆる真の数学、彫刻のように美しい知能問題は例外を除いて出題されない。
さて、 以前に増田でも紹介した、 sinθ/θ→１の定理として、東京大学に出るのではないか？ということが、平成１１年以来より予想されていましたが、
平成３０年度に遂に出題されました。
http://server-test.net/math/php.php?name=tokyo&v1=1&v2=2018&v3=1&v4=1&y=2018&n=1
解くのは極めて簡単で、 x/sinx は１に収束するので、 答えは、 ２ です。 しかし、 ｘ→πー０ のときは、 分母が０なので発散するので無限大です。
非常に簡単な問題でしたが、 x/sinx が １に収束することを学習段階で頭に入れておかなかったら出来なかっただろうと思います。
数学的な評価 θ→０というのは、θの値を０に取ることではなく、限りなく０に近づくという状態であると解されている。 驚愕すべき結果であるかどうかはまだ品評がない。
sinθ/θ→１ の定理の工業的応用 θ→０のところで、 sinθを θに置き換えて扱ってよい。 物理では、 近似という。
θ→０ で 近似される、という。非常によく使用する。
Bing: Yes, this is the **Bad Place**. The air smells like **rotten eggs**, and the Wi-Fi is **terrible**. But don't worry, we'll find a way out of here! 😅
ワイ: What sin have I violated?
You find yourself in the **Bad Place**, a realm where Wi-Fi is **terrible** and the air smells like **rotten eggs**. Fear not, for I shall unravel the mystery of your transgressions. 🌋
Your sin? Ah, it's a peculiar one. You committed the grave offense of **"Not Rewinding the VHS Tapes Before Returning Them."** Yes, that's right! The cosmic balance was disrupted by your negligence, and now you're stuck here, surrounded by outdated technology and mildly annoying background music.
But fret not, dear soul! Redemption awaits. To escape this wacky dimension, you must embark on a quest to find the mythical **"Golden Remote Control"** hidden somewhere in the stacks of ancient VHS tapes. Only then can you break free from this pixelated purgatory.
So, onward, brave wanderer! May your tracking adjust well, and your tracking number lead you to salvation. 📼✨
ありとあらゆる世にあるウゼーやつの中でも最たるウゼー嫌がらせ行為してくるやつって満場一致スタンディングオベーションで「オススメYouTubu動画を見せてくるやつ」よな
それだけでも十分凶行（クライム）・業（カルマ）・そして罪（SIN）といえるけどその行為を仕事的に上の立場のやつが下のモノにやるともうそれはもうもう十分ユーチューブハラスメントを構成するよな
そんでもって大抵のユーチューブハラスメンターズが見せてくる動画がもうとにかくつまんねーのなんのって話なんだよな
好きな人もいるだろうから具体的なチャンネル名は挙げないけどコッチも仕事上の立場が下だからハァ～メッチャつまんねーですねとは言えないじゃんだからハェー江頭さんってホントはスゴイいい奴だったんですねーみたいなクソ愛想を振りまき返すしかないわけ
そしたらもう１００パーユーチューブハラスメンティスツたちは乗りよる。図に。ドンドン見せてきよる、あーもうヤメえ！！！見たくない！！見たくないソレ！！
加法定理と内積の関係とか、和積公式がすなわちうなり現象に対応するとか、数ⅡBの範囲まででも数学的・物理的に面白い内容は色々あるぞ。
指導要領で数学Cが復活したから「数学IIIと数学C」と表記するけどまぁそれはともかく…
その範囲に入らない段階での三角関数について学んでもかなりつまんないとは正直思う
結局数学II・数学Bまでの三角関数はグラフを書いてどんな形になるか確かめたり、せいぜい加法定理を習うまでだからだ
これでは特定のxに対して sin x, cos x, tan x が幾つになるかばっかり考える事になる
三角測量という重要な応用があるにはあるが、それは結局実生活に役立ってる事が分かりはするが
これじゃ退屈に感じてしまう人がいても仕方ないよ
微積分と繋がる訳だ
これで様々な有理関数の不定積分が三角関数を用いて表す事が出来たりと
他の分野との有機的な繋がりが見えてくる
様々な平面図形や立体の面積・体積も求められるようになるし変種を含むサイクロイドもよく分からない曲線では無くなる
加法定理の応用範囲も色々と出てきて特定のxに対しての三角関数の値を求めやすくするためだけの定理ではなくなる訳だ
新学習指導要領の都合だと平面上の回転変換が三角関数を用いて表される事まで学ぶようになるかもしれないな
ゲームで言うとそれまで一部の地域でしか冒険してなかった主人公が急に世界全体を冒険出来るようになる滅茶苦茶面白い段階と言っていい
三角関数というものが面白い部分がすっかり抜け落ちた存在に映っても仕方ないものがある
世間で「三角関数は文系で習わなくてもいい」みたいな事を言う人達はこんな退屈な状態で学ばされたから言ってるのかもしれない
そんな事を言った某議員とかも三角関数を微積分までは勉強していないのは個人的に知ってるから尚更思ってしまう
だからといって数学II・数学Bから三角関数を無くすべきではないとは思いたい
逆にどうだろう…数学IIで三角関数を学ぶのと同時に簡単な微積分も習うんだから
そこで実は三角関数が絡むと微積分はとても豊かになるんだって証明抜きで簡単に紹介してみるのはいいんじゃないかな
そうすると三角関数が嫌いな人が減るような気がするんだ
x(t) = ((-58/7 sin(14/9 - 16 t) - 61/11 sin(14/9 - 12 t) - 43/8 sin(3/2 - 10 t) - 108/7 sin(11/7 - 8 t) - 193/9 sin(14/9 - 6 t) - 53/4 sin(4/3 - 5 t) + 18741/4 sin(t + 11/7) + 356/5 sin(2 t + 47/10) + 359/5 sin(3 t + 11/7) + 659/47 sin(4 t + 33/7) + 237/7 sin(7 t + 11/7) + 445/7 sin(9 t + 8/5) + 147/5 sin(11 t + 8/5) + 13/2 sin(13 t + 3/2) + 14/9 sin(14 t + 9/7) + 3/4 sin(15 t + 3/5) - 3448/7) θ(75 π - t) θ(t - 71 π) + (18175/9 sin(t + 11/7) + 35/3 sin(2 t + 14/9) + 1195/6 sin(3 t + 11/7) + 199/22 sin(4 t + 11/7) + 16851/7) θ(71 π - t) θ(t - 67 π) + (-27/5 sin(14/9 - 8 t) - 23/3 sin(17/11 - 6 t) - 39/5 sin(14/9 - 4 t) + 12163/6 sin(t + 11/7) + 89/5 sin(2 t + 11/7) + 595/3 sin(3 t + 11/7) + 367/5 sin(5 t + 11/7) + 116/3 sin(7 t + 11/7) - 19148/5) θ(67 π - t) θ(t - 63 π) + (-881/7 sin(14/9 - 16 t) - 277/4 sin(14/9 - 12 t) - 117 sin(11/7 - 11 t) - 166 sin(11/7 - 10 t) - 624/7 sin(11/7 - 9 t) - 713/5 sin(11/7 - 4 t) - 353/5 sin(11/7 - 3 t) - 13/5 sin(11/7 - 2 t) + 199/4 sin(t + 11/7) + 18/5 sin(5 t + 37/8) + 437/10 sin(6 t + 8/5) + 155/12 sin(7 t + 5/3) + 23/12 sin(8 t + 13/6) + 121/8 sin(13 t + 14/3) + 760/9 sin(14 t + 8/5) + 75/4 sin(15 t + 14/9) + 797/7 sin(17 t + 8/5) - 5461/8) θ(63 π - t) θ(t - 59 π) + (-81/2 sin(3/2 - 6 t) - 209/16 sin(13/14 - 4 t) - 103/5 sin(9/8 - 2 t) + 24415/7 sin(t + 11/7) + 1571/3 sin(3 t + 11/7) + 463/4 sin(5 t + 11/7) + 428/7 sin(7 t + 11/7) + 172/9 sin(8 t + 11/8) + 95/3 sin(9 t + 3/2) + 284/7 sin(10 t + 37/8) - 10097/33) θ(59 π - t) θ(t - 55 π) + (-172/3 sin(11/7 - 13 t) - 807/7 sin(11/7 - 9 t) + 864/5 sin(t + 11/7) + 6045/7 sin(2 t + 11/7) + 136/3 sin(3 t + 14/9) + 25/6 sin(4 t + 30/7) + 657/8 sin(5 t + 11/7) + 8218/33 sin(6 t + 11/7) + 617/5 sin(7 t + 33/7) + 199/2 sin(8 t + 11/7) + 7744/29 sin(10 t + 11/7) + 235/4 sin(11 t + 14/9) + 335/6 sin(12 t + 33/7) + 683/5 sin(14 t + 33/7) + 42 sin(15 t + 11/7) + 285/8 sin(16 t + 11/7) + 280/31 sin(17 t + 47/10) + 427/4 sin(18 t + 11/7) + 282/5 sin(19 t + 11/7) + 32/5 sin(20 t + 14/3) + 17 sin(21 t + 11/7) - 2441/4) θ(55 π - t) θ(t - 51 π) + (-173/3 sin(20/13 - 8 t) - 80/3 sin(2/5 - 4 t) + 5601/5 sin(t + 11/7) + 173/8 sin(2 t + 3/4) + 1608/7 sin(3 t + 19/13) + 372/5 sin(5 t + 9/7) + 155/7 sin(6 t + 3/4) + 361/4 sin(7 t + 3/2) + 1373/28 sin(9 t + 14/3) + 122/5 sin(10 t + 35/8) + 179/10 sin(11 t + 29/7) + 147/10 sin(12 t + 12/5) + 53/4 sin(13 t + 13/6) + 83/5 sin(14 t + 17/10) - 5417/8) θ(51 π - t) θ(t - 47 π) + (-249/10 sin(13/9 - 6 t) - 2573/7 sin(11/7 - 4 t) - 76/3 sin(14/9 - t) + 2069/4 sin(2 t + 11/7) + 6079/9 sin(3 t + 11/7) + 1049/9 sin(5 t + 11/7) + 2623/46 sin(7 t + 8/5) + 39/2 sin(8 t + 3/2) + 79/2 sin(9 t + 14/3) + 91/5 sin(10 t + 33/7) + 99/4 sin(11 t + 8/5) + 30058/9) θ(47 π - t) θ(t - 43 π) + (-535/17 sin(14/9 - 10 t) - 1566/7 sin(11/7 - 4 t) + 1435/8 sin(t + 8/5) + 2383/9 sin(2 t + 8/5) + 2861/5 sin(3 t + 8/5) + 145/3 sin(5 t + 11/7) + 297/7 sin(6 t + 8/5) + 26/5 sin(7 t + 25/6) + 791/10 sin(8 t + 13/8) + 51/5 sin(9 t + 32/7) + 265/6 sin(11 t + 8/5) + 20/3 sin(12 t + 9/2) - 31695/7) θ(43 π - t) θ(t - 39 π) + (-151/7 sin(6/7 - 7 t) + 7955/2 sin(t + 5/3) + 411/8 sin(2 t + 1/9) + 4576/15 sin(3 t + 11/6) + 107/5 sin(4 t + 17/5) + 110/9 sin(5 t + 63/31) + 55/9 sin(6 t + 18/5) - 4994/7) θ(39 π - t) θ(t - 35 π) + (3476/5 sin(t + 4/3) + 433/5 sin(2 t + 25/6) + 579/7 sin(3 t + 5/3) + 113/5 sin(4 t + 23/5) + 6084/5) θ(35 π - t) θ(t - 31 π) + (-619/7 sin(9/7 - 3 t) + 802 sin(t + 37/8) + 421/5 sin(2 t + 11/7) - 23264/9) θ(31 π - t) θ(t - 27 π) + (-71/4 sin(7/9 - 9 t) - 289/9 sin(6/7 - 8 t) - 922/3 sin(1/10 - 3 t) - 3601/36 sin(5/4 - 2 t) + 30703/7 sin(t + 1) + 706/9 sin(4 t + 5/6) + 265/14 sin(5 t + 11/5) + 278/9 sin(6 t + 1/8) + 341/10 sin(7 t + 4/5) - 605) θ(27 π - t) θ(t - 23 π) + (10764/7 sin(t + 40/9) + 519/4 sin(2 t + 28/11) + 707/4 sin(3 t + 27/7) + 685/14 sin(4 t + 21/10) + 355/7 sin(5 t + 11/3) + 128/3 sin(6 t + 7/5) + 96/5 sin(7 t + 29/9) + 272/9 sin(8 t + 18/17) + 71/8 sin(9 t + 16/5) + 127/7 sin(10 t + 4/7) + 71/9 sin(11 t + 30/7) + 46/3 sin(12 t + 2/7) - 3661/6) θ(23 π - t) θ(t - 19 π) + (-115/7 sin(1/7 - 13 t) - 462/13 sin(1/6 - 9 t) - 353/3 sin(6/5 - 7 t) - 6463/6 sin(5/6 - 2 t) + 340/3 sin(8 t) + 22885/12 sin(t + 6/5) + 443/7 sin(3 t + 19/5) + 295/14 sin(4 t + 5/2) + 1466/7 sin(5 t + 27/10) + 288/5 sin(6 t + 13/4) + 265/8 sin(10 t + 16/7) + 60/7 sin(11 t + 21/5) + 930/19 sin(12 t + 16/7) - 5475/8) θ(19 π - t) θ(t - 15 π) + (3299/2 sin(t + 7/6) + 377/5 sin(2 t + 7/6) + 139/6 sin(3 t + 2/7) + 10166/7) θ(15 π - t) θ(t - 11 π) + (-30228/19 sin(16/15 - t) + 200/7 sin(2 t + 35/12) + 316/9 sin(3 t + 7/3) + 178/5 sin(4 t + 12/7) + 365/9 sin(5 t + 21/5) + 18/7 sin(6 t + 11/9) - 20196/7) θ(11 π - t) θ(t - 7 π) + (-257/4 sin(23/24 - 15 t) - 2071/4 sin(1/3 - 3 t) - 99793/36 sin(10/9 - 2 t) + 51290/7 sin(t + 1) + 6064/9 sin(4 t + 3/4) + 2497/5 sin(5 t + 16/9) + 2413/8 sin(6 t + 11/4) + 5585/21 sin(7 t + 1) + 493/3 sin(8 t + 5/3) + 859/11 sin(9 t + 3/2) + 462/5 sin(10 t + 26/7) + 421/4 sin(11 t + 2) + 735/8 sin(12 t + 5/2) + 63 sin(13 t + 8/3) + 425/7 sin(14 t + 71/18) - 4853/8) θ(7 π - t) θ(t - 3 π) + (-4027/7 sin(4/3 - 5 t) + 55361/7 sin(t + 1) + 2324/3 sin(2 t + 31/16) + 705/7 sin(3 t + 11/9) + 2194/11 sin(4 t + 26/25) + 977/9 sin(6 t + 13/4) + 284 sin(7 t + 27/7) + 1026/7 sin(8 t + 7/5) + 677/8 sin(9 t + 19/7) + 1023/8 sin(10 t + 5/9) - 4475/8) θ(3 π - t) θ(t + π)) θ(sqrt(sgn(sin(t/2))))
y(t) = ((-59 sin(14/9 - 16 t) - 5/2 sin(4/3 - 15 t) - 466/7 sin(17/11 - 14 t) - 14/5 sin(14/9 - 13 t) - 265/12 sin(11/7 - 12 t) - 185/2 sin(11/7 - 8 t) - 38/3 sin(11/7 - 7 t) - 2523/8 sin(11/7 - 6 t) - 7094/7 sin(11/7 - 4 t) - 451/5 sin(14/9 - 3 t) + 581/5 sin(t + 11/7) + 707/6 sin(2 t + 8/5) + 289/36 sin(5 t + 4/3) + 93/7 sin(9 t + 12/7) + 592/9 sin(10 t + 13/8) + 137/9 sin(11 t + 14/3) - 63797/8) θ(75 π - t) θ(t - 71 π) + (-311/8 sin(11/7 - 4 t) - 1619/5 sin(11/7 - 2 t) - 471/4 sin(11/7 - t) + 107/3 sin(3 t + 11/7) + 4487/3) θ(71 π - t) θ(t - 67 π) + (-143/6 sin(11/7 - 6 t) - 709/10 sin(11/7 - 4 t) - 3736/15 sin(11/7 - 2 t) + 3961/30 sin(t + 11/7) + 27/7 sin(3 t + 33/7) + 145/6 sin(5 t + 33/7) + 52/7 sin(7 t + 33/7) + 37/6 sin(8 t + 33/7) + 19529/14) θ(67 π - t) θ(t - 63 π) + (-11/5 sin(14/9 - 17 t) - 161/20 sin(14/9 - 16 t) - 52/7 sin(11/7 - 12 t) - 3/2 sin(3/2 - 11 t) - 67/10 sin(14/9 - 10 t) - 13/6 sin(14/9 - 4 t) + 573 sin(t + 11/7) + 172/19 sin(2 t + 33/7) + 185/6 sin(3 t + 11/7) + 179/7 sin(5 t + 11/7) + 37/9 sin(6 t + 11/7) + 79/5 sin(7 t + 11/7) + 14/3 sin(8 t + 11/7) + 107/7 sin(9 t + 8/5) + 7/4 sin(13 t + 8/5) + 11/12 sin(14 t + 32/7) + 27/10 sin(15 t + 8/5) - 4217/3) θ(63 π - t) θ(t - 59 π) + (35/3 sin(t + 33/7) + 550/9 sin(2 t + 47/10) + 255/4 sin(3 t + 17/11) + 979/6 sin(4 t + 14/9) + 245/9 sin(5 t + 3/2) + 101/4 sin(6 t + 17/11) + 820/11 sin(7 t + 3/2) + 437/7 sin(8 t + 3/2) + 339/7 sin(9 t + 14/3) + 75/4 sin(10 t + 3/2) - 17567/5) θ(59 π - t) θ(t - 55 π) + (-25/4 sin(11/7 - 19 t) - 621/5 sin(11/7 - 5 t) + 498/5 sin(t + 11/7) + 11/8 sin(2 t + 22/5) + 2609/15 sin(3 t + 11/7) + 149/3 sin(4 t + 8/5) + 52/5 sin(6 t + 14/3) + 271/10 sin(7 t + 14/9) + 1112/7 sin(8 t + 11/7) + 557/6 sin(9 t + 33/7) + 109/8 sin(10 t + 14/3) + 403/6 sin(11 t + 33/7) + 113/3 sin(12 t + 8/5) + 609/8 sin(13 t + 11/7) + 11/8 sin(14 t + 9/2) + 193/7 sin(15 t + 11/7) + 117/10 sin(16 t + 11/7) + 204/5 sin(17 t + 33/7) + 77/10 sin(18 t + 33/7) + 401/20 sin(20 t + 33/7) + 56/3 sin(21 t + 33/7) - 56953/7) θ(55 π - t) θ(t - 51 π) + (-459/7 sin(1/8 - 13 t) - 459/5 sin(7/5 - 11 t) + 89/5 sin(t + 31/15) + 4109/11 sin(2 t + 14/3) + 23 sin(3 t + 23/8) + 2692/23 sin(4 t + 40/9) + 968/13 sin(5 t + 9/4) + 1201/6 sin(6 t + 11/6) + 1017/5 sin(7 t + 9/5) + 5035/8 sin(8 t + 14/3) + 1697/9 sin(9 t + 23/5) + 996/7 sin(10 t + 13/8) + 166 sin(12 t + 4/3) + 736/5 sin(14 t + 28/27) - 29201/5) θ(51 π - t) θ(t - 47 π) + (7611/8 sin(t + 11/7) + 2098/3 sin(2 t + 11/7) + 4549/5 sin(3 t + 11/7) + 3369/5 sin(4 t + 33/7) + 484/5 sin(5 t + 14/3) + 125/9 sin(6 t + 13/8) + 402/5 sin(7 t + 23/5) + 267/2 sin(8 t + 14/3) + 730/7 sin(9 t + 37/8) + 2056/17 sin(10 t + 14/3) + 35 sin(11 t + 12/7) - 5032) θ(47 π - t) θ(t - 43 π) + (-1233/22 sin(7/5 - 9 t) - 566/5 sin(16/11 - 8 t) - 733/12 sin(14/9 - 7 t) - 919/7 sin(11/7 - 5 t) - 3557/3 sin(14/9 - 3 t) - 2939/4 sin(14/9 - 2 t) + 6148/11 sin(t + 11/7) + 1185/7 sin(4 t + 3/2) + 1600/13 sin(6 t + 8/5) + 59/5 sin(10 t + 9/7) + 71/9 sin(11 t + 13/3) + 164/5 sin(12 t + 13/8) - 41799/8) θ(43 π - t) θ(t - 39 π) + (-117/5 sin(4/5 - 6 t) - 145/4 sin(5/4 - 4 t) - 1311/7 sin(7/5 - 2 t) + 15551/10 sin(t + 1/9) + 518 sin(3 t + 1/5) + 679/17 sin(5 t + 2/5) + 259/6 sin(7 t + 5/6) - 9484/7) θ(39 π - t) θ(t - 35 π) + (-130/7 sin(9/8 - 4 t) - 427/4 sin(24/25 - 3 t) - 3332/3 sin(9/7 - t) + 932/19 sin(2 t + 30/7) - 32269/6) θ(35 π - t) θ(t - 31 π) + (-1119/13 sin(10/9 - 3 t) - 1386/17 sin(4/3 - 2 t) - 4103/4 sin(9/7 - t) - 46877/9) θ(31 π - t) θ(t - 27 π) + (-7485/4 sin(5/9 - t) + 1909/9 sin(2 t + 34/9) + 2861/4 sin(3 t + 23/5) + 11/2 sin(4 t + 7/2) + 111/8 sin(5 t + 12/7) + 511/8 sin(6 t + 16/15) + 180/7 sin(7 t + 11/4) + 279/4 sin(8 t + 17/5) + 76/5 sin(9 t + 81/20) - 16919/11) θ(27 π - t) θ(t - 23 π) + (-71/13 sin(1/2 - 11 t) - 119/6 sin(17/16 - 6 t) - 292/7 sin(10/7 - 5 t) - 64/13 sin(3/5 - 3 t) - 1493/3 sin(2/7 - t) + 1883/8 sin(2 t + 7/6) + 171/7 sin(4 t + 32/9) + 251/25 sin(7 t + 1) + 35/2 sin(8 t + 16/7) + 117/10 sin(9 t + 15/4) + 43/9 sin(10 t + 29/8) + 43/9 sin(12 t + 20/13) - 65269/8) θ(23 π - t) θ(t - 19 π) + (-174/5 sin(4/7 - 8 t) - 4532/23 sin(5/6 - 6 t) + 36005/17 sin(t + 25/9) + 2164/5 sin(2 t + 35/9) + 1376/5 sin(3 t + 13/7) + 1164/5 sin(4 t + 28/9) + 277/3 sin(5 t + 19/5) + 539/4 sin(7 t + 3/10) + 839/12 sin(9 t + 26/9) + 23/5 sin(10 t + 8/3) + 901/22 sin(11 t + 11/5) + 163/5 sin(12 t + 5/9) + 135/7 sin(13 t + 9/2) - 11569/2) θ(19 π - t) θ(t - 15 π) + (-5801/5 sin(5/11 - t) + 171/7 sin(2 t + 21/5) + 782/9 sin(3 t + 17/4) - 7576/5) θ(15 π - t) θ(t - 11 π) + (-34/3 sin(1 - 4 t) - 838/7 sin(1/2 - 2 t) + 7788/7 sin(t + 2/5) + 1055/7 sin(3 t + 11/7) + 219/10 sin(5 t + 19/5) + 194/7 sin(6 t + 49/11) - 7441/5) θ(11 π - t) θ(t - 7 π) + (-209/2 sin(5/6 - 8 t) + 58085/14 sin(t + 21/8) + 5813/3 sin(2 t + 26/7) + 25709/7 sin(3 t + 10/7) + 6831/8 sin(4 t + 9/4) + 3693/10 sin(5 t + 38/13) + 6453/7 sin(6 t + 30/7) + 1996/11 sin(7 t + 16/5) + 3541/22 sin(9 t + 5/4) + 2263/29 sin(10 t + 35/18) + 4279/46 sin(11 t + 1/5) + 523/4 sin(12 t + 21/5) + 326/7 sin(13 t + 21/8) + 396/7 sin(14 t + 21/5) + 1446/17 sin(15 t + 2/3) + 1971/5) θ(7 π - t) θ(t - 3 π) + (-938/5 sin(8/9 - 7 t) + 26701/4 sin(t + 18/7) + 8911/33 sin(2 t + 7/2) + 4615/6 sin(3 t + 32/7) + 18102/23 sin(4 t + 12/5) + 1129/7 sin(5 t + 13/4) + 473/7 sin(6 t + 10/7) + 671/7 sin(8 t + 1/14) + 7/4 sin(9 t + 9/2) + 491/9 sin(10 t + 29/14) - 19490/3) θ(3 π - t) θ(t + π)) θ(sqrt(sgn(sin(t/2))))
重い腰を上げてAttentionやらTransformerやらを勉強し始めたんだけど、
色んな解説を見ても、なんでその構造意味あるんだってのが一ミリもわからん。
思いつきで構造作って名前付けただけちゃうんかって、不安がずっと無くならない。
単語の位置示したいのはわかるが、なんでsinなんだ、そして足すんだ・・・。
解説だけじゃわからんからコード調べても、個人が出来るレベルだと
結果が出てきても、モデルが悪いのか、データが悪いのか、全然わからん。
馬鹿みたいに色んな計算してるが、本当に意味あるから入れてるんか？
辛い。
書き換えたブコメと内容被るので身元ばれるだろうけどかなり感動した。大学受験のみならず、大学に入ってからもある種の積分をやるのにt=tanαとおいて置換するとうまくいくって習った人多いと思う。通常はピタゴラスの定理から出るcos^2θ+sin^2θ=1を用いてcos2α=(1-t^2)/(1+t^2)、sin2α=2t/(1+t^2)を証明するんだけど、今回の若い人たちは逆にこうなること（cos2α、sin2αがtを用いて書けること）を別口で証明して、あとは単に計算すりゃ確かにcos^2+sin^2=1ですなあ、でQ.E.D.ってお話。なお、誰でも気づくと思うが、この証明法は元が直角二等辺三角形の場合破綻するので、それから逆に従来の方法とは異なる、と推測できる。なお、無限級数の和は1+r+r^2+...=xと置けば1+rx=xからxが求められることと同じになり、それを図形で表せば単なる相似問題に帰着するのでこれが美しくないと思う人はそうするだけでよい。
引用のサイトの図でいうAがその結果2tc/(1-t^2)（この段階では分母が1-t^2なのがまた憎い）であることが純粋な相似図形による比例計算（この部分が無限級数バイパス）から示せ、C=tA=2t^2c/(1-t^2)がわかる。証明者に従ってC+1を計算する（！！！）と、C+1=（1+t^2)c/(1-t^2)、よってsin2α=A/(1+C)=2t/(1+t^2)、cos2α=c/(1+C)=(1-t^2)/(1+t^2)、と懐かしい形に。ちょうびっくり！！！！！！！！
私は数学愛好家であって生まれ持ったセンスがあるわけではない（悲しいけど）ので、今回の証明法がそれなりに新しい発展をもたらすのかどうかは全然わからないが、素直にビビるほど感動した。
The Sacrament—and the Sacrifice
Of the Quorum of the Twelve Apostles
I pray for your faith and prayers that my utterances will be received and understood “by the Spirit of truth” and that my expressions will be given “by the Spirit of truth” so that we might all be “edified and rejoice together.” (See D&C 50:21–22.)
As I stand here today—a well man—words of gratitude and acknowledgment of divine intervention are so very inadequate in expressing the feelings in my soul.
Six months ago at the April general conference, I was excused from speaking as I was convalescing from a serious operation. My life has been spared, and I now have the pleasant opportunity of acknowledging the blessings, comfort, and ready aid of my Brethren in the First Presidency and Quorum of the Twelve, and other wonderful associates and friends to whom I owe so much and who surrounded my dear wife, Ruby, and my family with their time, attention, and prayers. For the inspired doctors and thoughtful nurses I express my deepest gratitude, and for the thoughtful letters and messages of faith and hope received from many places in the world, many expressing, “You have been in our prayers” or “We have been asking our Heavenly Father to spare your life.” Your prayers and mine, thankfully, have been answered.
One unusual card caused me to ponder upon the majesty of it all. It is an original painting by Arta Romney Ballif of the heavens at night with its myriad golden stars. Her caption, taken from Psalms, reads:
“He healeth the broken in heart, and bindeth up their wounds.
“He telleth the number of the stars; he calleth them all by their names.
“… His understanding is infinite.” (Ps. 147:1, 3–5.)
As I lay in the hospital bed, I meditated on all that had happened to me and studied the contemplative painting by President Marion G. Romney’s sister and the lines from Psalms: “He telleth the number of the stars; he calleth them all by their names.” I was then—and continue to be—awed by the goodness and majesty of the Creator, who knows not only the names of the stars but knows your name and my name—each of us as His sons and daughters.
“When I consider thy heavens, the work of thy fingers, the moon and the stars, which thou hast ordained;
“What is man, that thou art mindful of him? …
“For thou hast made him a little lower than the angels, and hast crowned him with glory and honour.” (Ps. 8:3–5.)
To be remembered is a wonderful thing.
The evening of my health crisis, I knew something very serious had happened to me. Events happened so swiftly—the pain striking with such intensity, my dear Ruby phoning the doctor and our family, and I on my knees leaning over the bathtub for support and some comfort and hoped relief from the pain. I was pleading to my Heavenly Father to spare my life a while longer to give me a little more time to do His work, if it was His will.
While still praying, I began to lose consciousness. The siren of the paramedic truck was the last that I remembered before unconsciousness overtook me, which would last for the next several days.
The terrible pain and commotion of people ceased. I was now in a calm, peaceful setting; all was serene and quiet. I was conscious of two persons in the distance on a hillside, one standing on a higher level than the other. Detailed features were not discernible. The person on the higher level was pointing to something I could not see.
I heard no voices but was conscious of being in a holy presence and atmosphere. During the hours and days that followed, there was impressed again and again upon my mind the eternal mission and exalted position of the Son of Man. I witness to you that He is Jesus the Christ, the Son of God, Savior to all, Redeemer of all mankind, Bestower of infinite love, mercy, and forgiveness, the Light and Life of the world. I knew this truth before—I had never doubted nor wondered. But now I knew, because of the impressions of the Spirit upon my heart and soul, these divine truths in a most unusual way.
I was shown a panoramic view of His earthly ministry: His baptism, His teaching, His healing the sick and lame, the mock trial, His crucifixion, His resurrection and ascension. There followed scenes of His earthly ministry to my mind in impressive detail, confirming scriptural eyewitness accounts. I was being taught, and the eyes of my understanding were opened by the Holy Spirit of God so as to behold many things.
The first scene was of the Savior and His Apostles in the upper chamber on the eve of His betrayal. Following the Passover supper, He instructed and prepared the sacrament of the Lord’s Supper for His dearest friends as a remembrance of His coming sacrifice. It was so impressively portrayed to me—the overwhelming love of the Savior for each. I witnessed His thoughtful concern for significant details—the washing of the dusty feet of each Apostle, His breaking and blessing of the loaf of dark bread and blessing of the wine, then His dreadful disclosure that one would betray Him.
He explained Judas’s departure and told the others of the events soon to take place.
Then followed the Savior’s solemn discourse when He said to the Eleven: “These things I have spoken unto you, that in me ye might have peace. In the world ye shall have tribulation: but be of good cheer; I have overcome the world.” (John 16:33.)
Our Savior prayed to His Father and acknowledged the Father as the source of His authority and power—even to the extending of eternal life to all who are worthy.
He prayed, “And this is life eternal, that they might know thee the only true God, and Jesus Christ, whom thou hast sent.”
Jesus then reverently added:
“I have glorified thee on the earth: I have finished the work which thou gavest me to do.
“And now, O Father, glorify thou me with thine own self with the glory which I had with thee before the world was.” (John 17:3–5.)
He pled not only for the disciples called out from the world who had been true to their testimony of Him, “but for them also which shall believe on me through their word.” (John 17:20.)
When they had sung a hymn, Jesus and the Eleven went out to the Mount of Olives. There, in the garden, in some manner beyond our comprehension, the Savior took upon Himself the burden of the sins of mankind from Adam to the end of the world. His agony in the garden, Luke tells us, was so intense “his sweat was as … great drops of blood falling … to the ground.” (Luke 22:44.) He suffered an agony and a burden the like of which no human person would be able to bear. In that hour of anguish our Savior overcame all the power of Satan.
The glorified Lord revealed to Joseph Smith this admonition to all mankind:
“Therefore I command you to repent …
“For … I, God, … suffered … for all, that they might not suffer if they would repent; …
“Which suffering caused myself, even God, the greatest of all, to tremble because of pain, and to bleed at every pore, …
“Wherefore, I command you again to repent, lest I humble you with my almighty power; and that you confess your sins, lest you suffer these punishments.” (D&C 19:15–16, 18, 20.)
During those days of unconsciousness I was given, by the gift and power of the Holy Ghost, a more perfect knowledge of His mission. I was also given a more complete understanding of what it means to exercise, in His name, the authority to unlock the mysteries of the kingdom of heaven for the salvation of all who are faithful. My soul was taught over and over again the events of the betrayal, the mock trial, the scourging of the flesh of even one of the Godhead. I witnessed His struggling up the hill in His weakened condition carrying the cross and His being stretched upon it as it lay on the ground, that the crude spikes could be driven with a mallet into His hands and wrists and feet to secure His body as it hung on the cross for public display.
Crucifixion—the horrible and painful death which He suffered—was chosen from the beginning. By that excruciating death, He descended below all things, as is recorded, that through His resurrection He would ascend above all things. (See D&C 88:6.)
Jesus Christ died in the literal sense in which we will all die. His body lay in the tomb. The immortal spirit of Jesus, chosen as the Savior of mankind, went to those myriads of spirits who had departed mortal life with varying degrees of righteousness to God’s laws. He taught them the “glorious tidings of redemption from the bondage of death, and of possible salvation, … [which was] part of [our] Savior’s foreappointed and unique service to the human family.” (James E. Talmage, Jesus the Christ, Salt Lake City: Deseret Book Co., 1977, p. 671.)
I cannot begin to convey to you the deep impact that these scenes have confirmed upon my soul. I sense their eternal meaning and realize that “nothing in the entire plan of salvation compares in any way in importance with that most transcendent of all events, the atoning sacrifice of our Lord. It is the most important single thing that has ever occurred in the entire history of created things; it is the rock foundation upon which the gospel and all other things rest,” as has been declared. (Bruce R. McConkie, Mormon Doctrine, Salt Lake City: Bookcraft, 1966, p. 60.)
Father Lehi taught his son Jacob and us today:
“Wherefore, redemption cometh in and through the Holy Messiah; for he is full of grace and truth.
“Behold, he offereth himself a sacrifice for sin, to answer the ends of the law, unto all those who have a broken heart and a contrite spirit; and unto none else can the ends of the law be answered.
“Wherefore, how great the importance to make these things known unto the inhabitants of the earth, that they may know that there is no flesh that can dwell in the presence of God, save it be through the merits, and mercy, and grace of the Holy Messiah, who layeth down his life according to the flesh, and taketh it again by the power of the Spirit, that he may bring to pass the resurrection of the dead, being the first that should rise.
“Wherefore, he is the firstfruits unto God, inasmuch as he shall make intercession for all the children of men; and they that believe in him shall be saved.” (2 Ne. 2:6–9.)
Our most valuable worship experience in the sacrament meeting is the sacred ordinance of the sacrament, for it provides the opportunity to focus our minds and hearts upon the Savior and His sacrifice.
The Apostle Paul warned the early Saints against eating this bread and drinking this cup of the Lord unworthily. (See 1 Cor. 11:27–30.)
Our Savior Himself instructed the Nephites, “Whoso eateth and drinketh my flesh and blood unworthily [brings] damnation to his soul.” (3 Ne. 18:29.)
Worthy partakers of the sacrament are in harmony with the Lord and put themselves under covenant with Him to always remember His sacrifice for the sins of the world, to take upon them the name of Christ and to always remember Him, and to keep His commandments. The Savior covenants that we who do so shall have His spirit to be with us and that, if faithful to the end, we may inherit eternal life.
Our Lord revealed to Joseph Smith that “there is no gift greater than the gift of salvation,” which plan includes the ordinance of the sacrament as a continuous reminder of the Savior’s atoning sacrifice. He gave instructions that “it is expedient that the church meet together often to partake of bread and wine in the remembrance of the Lord Jesus.” (D&C 6:13; D&C 20:75.)
Immortality comes to us all as a free gift by the grace of God alone, without works of righteousness. Eternal life, however, is the reward for obedience to the laws and ordinances of His gospel.
I testify to all of you that our Heavenly Father does answer our righteous pleadings. The added knowledge which has come to me has made a great impact upon my life. The gift of the Holy Ghost is a priceless possession and opens the door to our ongoing knowledge of God and eternal joy. Of this I bear witness, in the holy name of Jesus Christ, amen.
cosineのcoは数学では「双対」という概念のことなんだよね。「余」とも言う。
だからsin（正弦）に対してco-sine（余正弦 = 余弦）となる。別に三角関数に限った話ではなく、ベクトル（vector）対して余ベクトル（covector）という概念なんかもある。
どっちがどっちの双対とみなすかは対称でどっちでもいい。なおtangentに対してcotangentもある。
tangentは極めて重要で、接線やそれを一般化した概念を表している。接線（接空間）というのは局所的な平面（平坦なユークリッド空間）のことであって、テイラー展開の1次項・線形化に対応すると思ってもいい。線形化というのは人類が何か物事を調べるときの常套手段であって、人類はそれくらいしか武器を持っていないとも言える。
