cosine

My Favourite HoTS Hanzo quotes 👌

“It is not enough that I should succeed. Others should fail”

“When in doubt, put a bow on it”

“Every year I sneak into Shimada castle to honour my fallen brother, you’d think at this point the clan would just close the gate”

“Revenge is never a straight line, it’s a forest. And like a forest it can be most enjoyable”

“Sine of x equals cosine of pi over 2 minus x, simple trigonometry”

“They say that ‘perfect’ is the enemy of good. So i guess I am a villain after all… *Haughty laughter*”

“Crossbows…*scoff*, toys for children”

“Either be silent OR START MAKING SENSE”

(The rest of them)

The simple harmonic oscillator

Anonymous asked: Please explain the intuition of solving the SHM equation.

Okay Anon! Here you go, this is my rendition.


The problem

You have a mass suspended on a spring. We want to know where the mass will be at any instant of time.

Describe the motion of the mass


The physical solution

Now before we get on to the math, let us first visualize the motion by attaching a spray paint bottle as the mass.

Oh, wait that seems like a function that we are familiar with - The sinusoid.

Without even having to write down a single equation, we have found out the solution to our problem. The motion that is traced  by the mass is a sinusoid.


But what do I mean by a sinusoid ?

If you took the plotted paper and tried to create that function with the help of sum of polynomials i.e x, x2, x3 … Now you this what it would like :

By taking an infinite of these polynomial sums you get the function Since this series of polynomial occurs a lot, its given the name - sine.

I hope this shed some light on the intuition of the SHM equation. Have fun!

10

On any given day, I analyze the binomial levels air displacement, friction and velocity. And compute over ten thousand calculations by cosine, square root and lately analytic geometry. By hand. There are twenty, bright, highly capable negro women in the west computing group, and we’re proud to be doing our part for the country. So yes, they let women do some things at NASA, Mr. Johnson. And it’s not because we wear skirts. It’s because we wear glasses. Have a good day.

Hidden Figures, 2016

independent.co.uk
A 3,700 year old Babylonian tablet rewrites the history of maths
“The Babylonians unique approach to arithmetic and geometry means this is not only the world’s oldest trigonometric table, it’s also the only completely accurate trigonometric table on record,” he said.

The Ancient Babylonians knew about a form of trigonometry more advanced than the modern-day version – about 1,000 years before its supposed invention by the Ancient Greeks, academics in Australia say.

The astonishing claim is based on a 3,700-year-old clay tablet inscribed with a table of numbers.

Known as Plimpton 322, it is already known to contain evidence that the Babylonians knew Pythagoras’ famous equation for right-angled triangles, long before the Greek philosopher gave his name to it.

And researchers at the University of New South Wales (UNSW) have claimed it also shows the Babylonians developed a highly sophisticated form of trigonometry – the system of maths used to describe angles that has tortured generations of school pupils with sine, cosine and tangent.

Continue Reading.

6

I will have you know, I was the first negro female student at West Virginia university graduate school. On any given day, I analyze the binomial levels air displacement, friction and velocity. And compute over ten thousand calculations by cosine, square root and lately analytic geometry. By hand. There are twenty, bright, highly capable negro women in the west computing group, and we’re proud to be doing our part for the country. So yes, they let women do some things at NASA, Mr. Johnson. And it’s not because we wear skirts. It’s because we wear glasses. Have a good day.

The exponential, sine and cosine functions in physics

Inspired from a math lecture…

From the point of view of a physicist, the exponential, sine and cosine functions are interesting only because they are the solutions to differential equations. To us, they are defined to be the solution to ODEs, akin to how the Hermite, Legendre, Laguerre etc. polynomials are defined to be solutions to PDEs.

P.S. If you ONLY knew about the geometric definition of the sine function, you’d have to be very smart to see that it’s the solution to d^x/dx^2=-(constant)x geometrically.

Think about it this way: let’s say you want to describe some natural phenomena you see, for example radioactive decay and simple harmonic motion. If you go about trying to solve the equations describing these phenomena, you would have discovered / invented the exponential and sine functions. Wow. The exponential and trig functions indeed are deep rooted in nature.

P.S.S. You can derive the sine and cosine functions yourself! Substitute a series solution into the simple harmonic motion equation and what you’ll find is the sum of the taylor expansions of sine and cosine.

2

day 51 of 100 days of productivity!
bearing in mind that exams are only temporary is really going to get me through tomorrow (full day consisting of absolutely killer subjects ahh). anyway i’m completely exhausted and this is from me helping my friend with multiple parts of maths!

The Promised No-study SAT Tips

I saw that a lot of you wanted these~ Disclaimer: You still have to know English and the basics of math for these. This goes especially if you’re not a native speaker - your English needs to be at a pretty good level.

General:

  1. Read. A lot. Whenever you see a text that’s at least a paragraph or two long, take time to practice skimming. If you’re bored and have a little time, take something, for example a food wrapper, and try to find occurrences of a word (for example “Acid” for food) as quickly as possible. Hard mode: look for synonyms.
  2. Practice filling out the answer sheet. This is a massive time-sink for a lot of people, so you should practice to eliminate it. Print out an example answer sheet and try filling out the circles quickly and accurately without distracting yourself a lot. Hard mode:Try doing it while not focusing only on the circles - look away or start thinking about the next question.
  3. Check. A lot. The main goal of this strategy is to leave yourself enough time when you’ve filled out an answer for each question when you’re calm, know the questions and can focus on checking. Try and go through the questions, thinking, “This question tests this and that.” If you have the time, look at each answer and identify the error in it (harder for the math questions, but loads of fun if you can do it).
  4. Think in patterns: Whenever you’re stuck on an example question, don’t just check the answer. Try and understand how the person found it, if this question is similar to others you have seen. The SAT only uses a few different types of questions, there will rarely be something to surprise you if you know the common patterns.
  5. Rest: The SAT is a very demanding exam. Give your brain time to relax - my advice would be not to do anything mentally strenuous the day before the test. Also, something I found out from competitions - bring chocolate. The sugar in it helps your brain work better and shrug off tiredness and eating it will draw blood away from your brain, effectively hibernating it for the break to conserve energy. Also, it’s just a really tasty snack!

Writing:

  1. Use the right format for the essay. There are a lot of easy points for using the four/five paragraph system. Introduction, Reason 1, Reason 2, Conclusion. Begin each paragraph with a topic sentence and follow up with a story from your life or a book/movie to illustrate it. This way, even without using fancy vocab or grammar, you can get the points for structure and critical thought. Now just try not to make any obvious spelling mistakes and call it a day!
  2. Try to quickly find an argument for the essay. They don’t actually rate how intelligent your argument is. So, take a minute or two, breathe deeply, and no matter how stupid your idea is, write it out. (You might still want to take caution with sensitive topics, especially if you’re an international. A dumb mistake I made in my first sitting was bashing on American charity - that definitely did not endear me to the proctors.)
  3. Paragraphs: You have to have experience reading - look at how the topic never changes abruptly. Insert sentences that link what’s written before and after the gap. Final sentences of paragraphs shouldn’t raise more questions.
  4. Sentence questions: Skim through the questions. Try to answer most of them, the first thing that comes to mind, and fill out the answer sheet immediately. Chances are, if it sounds good to you, it’s the correct choice. Do this quickly, then try and do the paragraphs. After you’ve done this, go back to the questions and start checking.
  5. They usually test for a few broad topics. Identify if each sentence fits one of the patterns and answer accordingly. For the others, try and think what error they might want you to make. If you know you have the time, look at each answer in turn and identify the mistake in it. The most common ways for you to change a sentence would be:
  • Fragments: Try and see if each clause has a subject and a verb. Example: “In the dim light, making his way through the cave.” -> “In the dim light, he makes his way through the cave.”
  • Subject-verb agreement: Make sure that the subject is the one actually doing the action and singular/plural match. Example: “Gathering stones, the river was blocked by the men.” Did the river gather stones? No.
  • Consistency: Make sure that something introduced one way is always referred to like that (don’t switch out ‘one’ for ‘you’ or ‘they’). Make sure there are no extra linkers (”Since I was there, but he went too.”). Check if any verbs change tense when they shouldn’t. Don’t compare apples to oranges (”His homework was as good as John.” -> “As good as John’s”).
  • Adverb or adjective? If it describes a verb, it has a ‘ly’. Example: “She winked playful.” -> “She winked playfully.”
  • Singular or plural? Make sure not to refer to a plural object in singular. “Pandas, numbering in the hundreds now, is an endangered species.”
  • Prepositions, linkers, all the small words Sadly, you’ll have to know how they’re used.

Reading

  1. Word fill: Note the answers that obviously don’t make sense. Mark the one of the others that sounds best to you (in the answer sheet, too!). If you don’t know one or more of the words, aim for simplicity. After you’ve quickly answered all of the reading questions, come back to these. Look at the relationships between the gap and the sentence - are you looking for a positive or negative word? Antonyms or synonyms to something before? Try and guess what unknown words mean. This way, you will probably be able to eliminate all the wrong answers.
  2. Reading comprehension: You are not tested for understanding the text. Keep this in mind. What you are actually trying to do here is quickly find synonyms. If the question asks for “Was Anna’s family a) warm b) cold c) the spawn of Cthulhu?”, chances are that the text contains “Anna’s relatives acted chilly.” or something like that. Read the first question. Skim the text until it comes to that topic, then look for synonyms of the answers. Don’t make deductions! If you come across a ‘general message’ or ‘tone of the author’ question, skip it and answer it at the end of the text. The other questions will be in the same order as the answers are mentioned in the text. Checking: If you have time, look at each answer and try to see what in the text could mislead somebody to make that mistake.

Mathematics

  1. Calculator use: My advice would be to not bring a complex graphing calculator. They just slow you down. Try and do most operations by hand, then use the calculator only for, well, calculations.
  2. Basic topics to know: You are expected to be familiar with how to rearrange equations (ab=1 is the same as a=1/b) and solve linear and quadratics; cosine and Pythagorean theorems; number representations of lines and their intersections; median, mean and mode.
  3. Solve like a crab! One of the best things I learnt in “Fun Math” classes was that problems are solved more easily if you work from the answer back. Try and see what you would need (in terms of information) to find the answer. Then look back to the text of the problem - is what you need there? In most SAT problems, it is, or you can easily find it.
  4. Visualise: Especially for distance or geometry problems, make a small chart of what’s happening. Make lines for the distances the cars traveled or draw that pesky cylinder. Try and see in your mind how different elements move and which stay the same.

I guess this is all that I can say for now. Of course, this is my strategy so it might not work for everyone or it might not work without practice, so don’t think it’s a miracle solve-all. I’m always open for questions about ideas or specific problems, just write an ask~ And good luck to all future test-takers!

Cheating - Reggie Mantle

SOOOOOO guess who does Riverdale stuff now. Me. Because when I get a new fandom, it’s all I think about for like two weeks.

Originally posted by rcggiemantle

In which the reader has the locker next to Archie, her best friend since kindergarten.

Words: 871

Warnings: None

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0:19 - Le porte dell'ascensore stanno per chiudersi, ma Dajana riesce a entrare per un soffio.

This is the place, sit down, you’re safe now.

Thom, sei l'amore ❤️