Speaker Dungsmear? Or?

McCarthy 2.0 seems a vast improvement over Speaker Dungsmear. So here’s the question. Was he once a conservative with a little fire in the belly that got captured by the system and now is finding the bellyfire again? Or is this all completely under duress?

For the moment, it doesn’t matter which one. But some day it will matter, and we will have our answer.

RINOs an Endangered Species?
If Only!

According to Wikipoo, et. al., the Northern White Rhinoceros (Ceratotherium simum cottoni) is a critically endangered species. Apparently two females live on a wildlife preserve in Sudan, and no males are known to be alive. So basically, this species is dead as soon as the females die of old age. Presently they are watched over by armed guards 24/7.

Biologists have been trying to cross them with the other subspecies, Southern White Rhinoceroses (Rhinoceri?) without success; and some genetic analyses suggest that perhaps they aren’t two subspecies at all, but two distinct species, which would make the whole project a lot more difficult.

I should hope if the American RINO (Parasitus rectum pseudoconservativum) is ever this endangered, there will be heroic efforts not to save the species, but rather to push the remainder off a cliff. Onto punji sticks. With feces smeared on them. Failing that a good bath in red fuming nitric acid will do.

But I’m not done ranting about RINOs.

The RINOs (if they are capable of any introspection whatsoever) probably wonder why they constantly have to deal with “populist” eruptions like the Trump-led MAGA movement. That would be because the so-called populists stand for absolutely nothing except for going along to get along. That allows the Left to drive the culture and politics.

Given the results of our most recent elections, the Left will now push harder, and the RINOs will now turn even squishier than they were before.

I well remember 1989-1990 in my state when the RINO establishment started preaching the message that a conservative simply couldn’t win in Colorado. Never mind the fact that Reagan had won the state TWICE (in 1984 bringing in a veto-proof state house and senate with him) and GHWB had won after (falsely!) assuring everyone that a vote for him was a vote for Reagan’s third term.

This is how the RINOs function. They push, push, push the line that only a “moderate” can get elected. Stomp them when they pull that shit. Tell everyone in ear shot that that’s exactly what the Left wants you to think, and oh-by-the-way-Mister-RINO if you’re in this party selling the same message as the Left…well, whythefuckexactly are you in this party, you lying piece of rancid weasel shit?

In Defense of Ranked Choice Voting

One of the biggest obstacles to direly-needed change is RINOs, and one of the weapons in their arsenal is the “Wasted Vote” argument.

Periodically a third party has arisen, trying to hold RINOs to account by putting pressure on them from outside of the party, since doing so from the inside has historically done very little good. But, even if you find a third party candidate who perfectly reflects your views, you’re likely to vote for the RINO anyway. Why? Because if you don’t, the Democrat might win, and that would be even worse. So if you vote for that third party (that few will vote for), you’re throwing your vote away and increasing the likelihood of the Democrat winning. (It’s half as much a gain for the Democrat, as actually voting for the Democrat would be. Not as much, but half as much. Because although you denied the R your vote, you did not flip your vote to the Democrat.)

The Republican Party Establishment knows you don’t love them. But they know you hate the Democrats worse, and they use that to continue to herd you into supporting them. With gritted teeth you cast your vote, but your vote counts the same whether you cast it enthusiastically. And the other alternative, pissing on the voting apparatus to express your actual feelings, is probably a felony.

But what if you could vote for that third party without increasing the chances of the Dem walking away with the prize?

This is what ranked choice voting, or instant runoff voting, can do provided it is properly implemented. (And this includes the votes, and only genuine votes, being counted honestly, of course. However, I’m going to compare it to what we have today, and pretend that is honestly done too. RCV can’t work if it’s not honestly administered, just like our current system isn’t working because it isn’t honestly administered.)

The idea behind RCV is to vote by expressing your order of preference. You could vote for the Patriot Party, then for the RINO Party as your second choice (and ignore the Democrat, the Green, the Overt Socialist Schmuckmonkey Party, etc).

What does this do? It nullifies the wasted vote argument. Your vote will be counted for the Patriot party, first, then instead of it being “wasted” when the Patriot Party loses, it ends up going to the RINO. Actually, it’s just barely possible that the Patriot Party would actually beat the RINO, if people weren’t all individually afraid to vote for it.

It’s just like the famous “Prisoner’s Dilemma” where your fear of other peoples’ actions prevents you from doing the optimal thing–and vice-versa. As long as Job Lowe is afraid to vote Patriot because he’s afraid you’ll vote RINO, you’ll have to vote RINO because you fear that Job Lowe will, because he fears you will.

So on the whole I like RCV. It gives you a no-risk way to vote against the RINO scum, and in favor of someone who deserves your vote.

The problem is, as done here in the US, it comes packaged with a “jungle primary.” A bunch of candidates get to put their name out there, and the top four (or so) candidates get onto the “main” ballot. This gives party establishments their way around the threat of a good third party bumping them off. Because they know that few people bother with primaries, and third parties don’t have the resources to run in a primary…so they throw two or three establishment hacks into the primary and they will probably beat the third party. The result is the RINOs end up with two of the four slots in the general election, and the Dems get the other two. Now there’s suddenly no third party candidate on the ballot at all.

If we were to combine RCV with the present system where each party could nominate exactly one candidate to appear on the November ballot, or at the very least, ensure minor parties could get onto the ballot with at least one candidate regardless of the primary, we would be getting somewhere, but the establishment is smarter than we like to give them credit for. They will support the jungle primary + RCV “solution” rather than the more appropriate one-candidate-per-party + RCV solution.

It’s not RCV that is the problem, it’s the primary structure grafted onto it.

Justice

It says “Justice” on the picture.

And I’m sure someone will post the standard joke about what the fish thinks about the situation.

But what is it?

Here’s a take, from a different context: It’s about how you do justice, not the justice that must be done to our massively corrupt government and media. You must properly identify the nature of a person, before you can do him justice.

Ayn Rand, On Justice (speaking through her character John Galt, in Atlas Shrugged):

Justice is the recognition of the fact that you cannot fake the character of men as you cannot fake the character of nature, that you must judge all men as conscientiously as you judge inanimate objects, with the same respect for truth, with the same incorruptible vision, by as pure and as rational a process of identification—that every man must be judged for what he is and treated accordingly, that just as you do not pay a higher price for a rusty chunk of scrap than for a piece of shining metal, so you do not value a rotter above a hero—that your moral appraisal is the coin paying men for their virtues or vices, and this payment demands of you as scrupulous an honor as you bring to financial transactions—that to withhold your contempt from men’s vices is an act of moral counterfeiting, and to withhold your admiration from their virtues is an act of moral embezzlement—that to place any other concern higher than justice is to devaluate your moral currency and defraud the good in favor of the evil, since only the good can lose by a default of justice and only the evil can profit—and that the bottom of the pit at the end of that road, the act of moral bankruptcy, is to punish men for their virtues and reward them for their vices, that that is the collapse to full depravity, the Black Mass of the worship of death, the dedication of your consciousness to the destruction of existence.

Ayn Rand identified seven virtues, chief among them rationality. The other six, including justice, she considered subsidiary because they are essentially different aspects and applications of rationality.

—Ayn Rand Lexicon (aynrandlexicon.com)

Justice Must Be Done.

Trump, it is supposed, had some documents.

Biden and company stole the country.

I’m sure enough of this that I put my money where my mouth is.

The prior election must be acknowledged as fraudulent, and steps must be taken to prosecute the fraudsters and restore integrity to the system. (This doesn’t necessarily include deposing Joe and Hoe and putting Trump where he belongs, but it would certainly be a lot easier to fix our broken electoral system with the right people in charge.)

Nothing else matters at this point. Talking about trying again in 2024 or 2026 is pointless otherwise. Which is not to say one must never talk about this, but rather that one must account for this in ones planning; if fixing the fraud in the system is not part of the plan, you have no plan.

This will necessarily be piecemeal, state by state, which is why I am encouraged by those states working to change their laws to alleviate the fraud both via computer and via bogus voters. If enough states do that we might end up with a working majority in Congress and that would be something Trump never really had.

Lawyer Appeasement Section

OK now for the fine print.

This is the WQTH Daily Thread. You know the drill. There’s no Poltical correctness, but civility is a requirement. There are Important Guidelines,  here, with an addendum on 20191110.

We have a new board – called The U Tree – where people can take each other to the woodshed without fear of censorship or moderation.

And remember Wheatie’s Rules:

1. No food fights
2. No running with scissors.
3. If you bring snacks, bring enough for everyone.
4. Zeroth rule of gun safety: Don’t let the government get your guns.
5. Rule one of gun safety: The gun is always loaded.
5a. If you actually want the gun to be loaded, like because you’re checking out a bump in the night, then it’s empty.
6. Rule two of gun safety: Never point the gun at anything you’re not willing to destroy.
7. Rule three: Keep your finger off the trigger until ready to fire.
8. Rule the fourth: Be sure of your target and what is behind it.

(Hmm a few extras seem to have crept in.)

Spot Prices

Last week:

Gold $1,943.60
Silver $23.73
Platinum $930.00
Palladium $1,290.00
Rhodium $4,750.00

This week, 3 PM MT on Friday, markets closed for the weekend

Gold $1,914.10
Silver $22.75
Platinum $921.00
Palladium $1,338.00
Rhodium $4,750.00

Everything down or flat except palladium. Interesting!

Sines and Cosines from Plain Old Arithmetic

Well after last week’s difficulty with editing I’m going to have another go at it.

I ended by stating that there was a way of computing the sine and cosine of any arbitrary angle, without knowing any other sines or cosines.

This is good because if you do need other sines and cosines as inputs, then roundoff error in the previously determined sines and cosines will start you off wrong, and you’ll be making an roundoff error here (remember you can’t specify most of these numbers precisely, they’re irrational). These errors will only accumulate. A direct method that doesn’t depend on other calculations would be much better.

We have that method. Sort of. There are a couple of caveats: This method will give you an approximation, though you can put more work into it and get it as close to the actual sine or cosine as you want. The more accurate you want, the more work you will need to do. (It becomes precisely accurate if you do infinite work.

And the other caveat is you have to give your angle in radians, because that is the “natural” way to measure an angle: as a distance along the arc of a circle.

OK, here we go:

Remember a few weeks ago (https://www.theqtree.com/2023/06/10/2023%C2%B706%C2%B710-joe-biden-didnt-win-daily-thread/) I talked about the Taylor series, and a special case of it called the MacLaurin series. These can be used, in some cases to compute values of a function that might otherwise seem intractable, and by using pure arithmetic.

The function must be infinitely differentiable, which is a calculus concept. I’m going to try to explain it, but if you don’t want to know it won’t hurt you not to know…much. You may want to skip the next few paragraphs…I’ll put bold where it’s safe to pick up so you can scan for it.

Basically, if you have a function like sinx which is shown below with x in radians.

…you can create another function that gives the slope of sin x at any given point. So what would such a function look like? Well, when x is zero, the slope of the wiggly line is just about 1. Which is to say, right there at x = 0, it’s climbing at the rate of one square vertically for every one unit to the right (in radians). Our new “slope of sin x function” has a value of 1 at x = 0. At π/2 (or 90°), the curve of sin x is level, so the slope is zero. At π, the slope is -1. At 3π/2, it’s zero again. And at 2π, it’s back to being 1 again.

We already know of a function that has these values at these points. It’s the cosine of x. Is that a coincidence? Actually, no. It turns out (after a lot more proof than I am going to show you), that the “slope of sin(x) function” is indeed cos(x). Now for some terminology; the correct way to phrase this is “the derivative of sin(x) is cos(x).”

You can repeat this exercise for cos(x), and you will get -sin(x). So the derivative of cos(x) is -sin(x). But cos(x) is already the derivative of sin(x), so the derivative of the derivative of sin(x) is -sin(x). But we don’t want to say “the derivative of the derivative of..” so we call that the second derivative.

Unsurprisingly, the derivative of -sin(x) is -cos(x). And the derivative of -cos(x) is back to sin(x). In other words, you can keep taking the derivative of the derivative of the derivative…forever. Kind of boring.

But that’s what it means to say that sin(x) is infinitely differentiable. With most functions you’ll eventually end up with a derivative that’s zero for all values of x, and then you’re done. But not with sin(x) and also (since it shows up in the chain) not for cos(x).

Which means that sine and cosine are infinitely differentiable and we can use the Maclaurin series on them.

(OK the calculus-averse are back with us now, and we can continue.)

So here is the Taylor series of any function, f. The idea is to pick a central point, a, and then compute the value of f for some nearby x. Note that all of the terms have x–a in them; you get closer if x is very close to a.

If a is zero, it becomes the Maclaurin series:

And the tick marks denote the first, second, third, and four derivatives.

The 1!, 2!, 3! etc are factorials. You get a factorial by multiplying the number by every smaller number down to 1. So 4! is 4 x 3 x 2 x 1. (Practically, of course, you don’t bother multiplying by 1.)

So if we set a equal to zero (we’re going to be interested in values for x close to zero), the formula can simplify.

I pointed out in the section you skipped over that the first derivative of sinx is cosx, the second derivative is -sinx, the third derivative is -cosx, the fourth derivative is sinx again, and so on…the cycle repeats.

So if I substitute those in we have:

sinx = sin(0) + (cos(0)/1!)x + (-sin(0)/2!)x² + (-cos(0)/3!)x³ + (sin(0)/4!)x⁴ + …

But we know that the sin(0) is zero, and so is -sin(0). So we can simply throw a bunch of these terms out (and as I do so I’ll bring in more terms from where the ellipsis was)

sinx = (cos(0)/1!)x +(-cos(0)/3!)x³ + (cos(0)/5!)x⁵ + (-cos(0)/7!)x⁷…

And the others (cos(0) and -cos(0)) can be replaced with 1 and -1 respectively, so simplifying:

sinx = x – x³/3! +x⁵/5! – x⁷/7!…

This is a nice pattern. Every odd power of x, divided by the factorial of the power. Every other term gets a minus sign. The further you carry it out, the more accurate you will be. Also, the closer to zero x is, the more accurate it will be, but you can always just carry it out further if x isn’t very close to zero, to make up for it. If you can follow this infinitely far, you can get the correct answers for numbers very far away from 0, as far away as you like.

Here is a graph of sinx, along with a graph of what you get out of the Maclaurin representation taken to the x⁷ term.

As a practical matter you do not need to go past x=π/2, because the function makes the mirror-image curve going down to π as it did climbing to π/2. So the sine of 89° is the same as the sine of 91°, and so on. And the stretch from π to 2π is also a mirror image, except flipped vertically. After that, it repeats. So even though you can see some difference at x = π, and a lot of difference at 2π, you can probably get everything you need from the range 0 to π/, where this is pretty close.

So you can get pretty close with this function, between –π/2 and π/2. I’ll repeat it with the factorials written out:

sinx is roughly equal to: x – x³/6 +x⁵/120 – x⁷/5040.

Certainly the brain inside a calculator should be able to do that very quickly! And so, that’s probably what it does. It might go out one or two more terms:

x – x³/6 +x⁵/120 – x⁷/5040 + x⁹/363,880 – x¹¹/39,916,800

But you can see if x is small you’re not going to get much change from the x¹¹ term.

So let’s do the same thing for cosines. Now when we go back to the MacLaurin series, everything shifts over, it starts with cosine, then -sine, then -cosine, then sine.

cosx = cos(0) + (-sin(0)/1!)x + (-cos(0)/2!)x² + (sin(0)/3!)x³ + (cos(0)/4!)x⁴ + …

And now we know that the cosine of zero is 1, and the sine of zero is zero, so:

cosx = 1 + –x²/2! + x⁴/4! + –x⁶/6! + x⁸/8! +…

So here we have a pattern oddly like, but oddly unlike, the sine version. Now it’s the even powers of x (including x⁰ = 1) that show up, again with alternating signs.

And we have:

cosx is roughly equal to: 1 – x²/2 +x⁴/24 – x⁶/720 + x⁸/40,320.

So we can get increasingly accurate. Because, remember sine and cosine swap at 45 degrees. The sine of 60° is the same as the cosine of 30°, so if you want the sine of 60°, compute the cosine of 30° instead; your answer will be closer. (Needless to say first, convert to radians!) So we really only need these series to be accurate enough for whatever our purposes are, to 45° or π/4 radians.

Next time: I pull a mathematical rabbit out of my hat!

Obligatory PSAs and Reminders

China is Lower than Whale Shit

To conclude: My standard Public Service Announcement. We don’t want to forget this!!!

Remember Hong Kong!!!

https://youtube.com/watch?v=L3tnH4FGbd0

中国是个混蛋 !!!
Zhōngguò shì gè hùndàn !!!
China is asshoe !!!

China is in the White House

Since Wednesday, January 20 at Noon EST, the bought-and-paid for Joseph Biden has been in the White House. It’s as good as having China in the Oval Office.

Joe Biden is Asshoe

China is in the White House, because Joe Biden is in the White House, and Joe Biden is identically equal to China. China is Asshoe. Therefore, Joe Biden is Asshoe.

But of course the much more important thing to realize:

Joe Biden Didn’t Win

乔*拜登没赢 !!!
Qiáo Bài dēng méi yíng !!!
Joe Biden didn’t win !!!

News Flash

Today, it is still the case that Joe Biden didn’t Win.

I realize that to some readers, this might be a shock; surely at some point things must change and Biden will have actually won.

But the past cannot actually be changed.

It will always and forever be the case that Joe Biden didn’t win.

And if you, Leftist Lurker, want to dismiss it as dead white cis-male logic…well, you can call it what you want, but then please just go fuck off. No one here buys that bullshit–logic is logic and facts are facts regardless of skin color–and if you gave it a moment’s rational thought, you wouldn’t either. Of course your worthless education never included being able to actually reason–or detect problems with false reasoning–so I don’t imagine you’ll actually wake up as opposed to being woke.

As Ayn Rand would sometimes point out: Yes, you are free to evade reality. What you cannot do is evade the consequences of evading reality. Or to put it concretely: You can ignore the Mack truck bearing down on you as you play in the middle of the street, you won’t be able to ignore the consequences of ignoring the Mack truck.

And Ayn Rand also pointed out that existence (i.e., the sum total of everything that exists) precedes consciousness–our consciousnesses are a part of existence, not outside of it–therefore reality cannot be a “social construct” as so many of you fucked-up-in-the-head people seem to think.

So much for Leftist douchebag lurkers. For the rest of you, the regular readers and those lurkers who understand such things: I continue to carry the banner once also carried by Wheatie. His Fraudulency didn’t win.

Let’s Go, Brandon!!

Hey China!

Or rather, “Hey Chinese Communist Party and your entire array of servitors, ass-wipers, and fellators!”

You’re not even worth my time this week. When you decide to act like civilized people, maybe I’ll give you a lesson or two in how non-barbarians behave.

Hey BiteMe!
(Or, Whoever Has Their Hand Rammed Up That Putrefying Meat Puppet’s Ass)

[Language warning]

You and yours have caused a lot of injury. Literal injury with your war on people who don’t want to take an untested vaccine. When people die in an emergency room because a hospital won’t admit them because they haven’t had their clot shot, that’s a crime.

I’m going to address here the insult on top of the injury, because I am among the insulted. I still have my health but apparently you want me to live under the 8th Street Bridge (which actually isn’t on 8th Street, but whatever, that’s what the I-25 overpass over Cimarron is called), so maybe if you have your way that won’t be true for long. Dreadful time of year to become homeless.

No, you’re just trying to make me unemployed, because I won’t take your fucking shots.

Well, that threat is NOT going to work. I. Won’t. Take. Your. Fucking. Shots.

And it looks like enough people agree, that you’re having to back down, you worthless asswipe.

You’re LOSING.

You LOSER.

You Chinese-bought ratfucking traitor.

I would love to see you die an agonizing, humiliating death. (This isn’t a threat, because I am not threatening to cause that death. I am just announcing my intention to party if it happens.) It would be just recompense for the way you’re killing America…and millions of Americans.

His Fraudulency

Joe Biteme, properly styled His Fraudulency, continues to infest the White House, we haven’t heard much from the person who should have been declared the victor, and hopium is still being dispensed even as our military appears to have joined the political establishment in knuckling under to the fraud.

One can hope that all is not as it seems.

I’d love to feast on that crow.

(I’d like to add, I find it entirely plausible, even likely, that His Fraudulency is also His Figureheadedness. (Apparently that wasn’t a word; it got a red underline. Well it is now.) Where I differ with the hopium addicts is on the subject of who is really in charge. It ain’t anyone we like.)

Justice Must Be Done.

The prior election must be acknowledged as fraudulent, and steps must be taken to prosecute the fraudsters and restore integrity to the system.

Nothing else matters at this point. Talking about trying again in 2024 is hopeless otherwise. Which is not to say one must never talk about this, but rather that one must account for this in ones planning; if fixing the fraud is not part of the plan, you have no plan.

Lawyer Appeasement Section

OK now for the fine print.

This is the WQTH Daily Thread. You know the drill. There’s no Poltical correctness, but civility is a requirement. There are Important Guidelines,  here, with an addendum on 20191110.

We have a new board – called The U Tree – where people can take each other to the woodshed without fear of censorship or moderation.

And remember Wheatie’s Rules:

1. No food fights
2. No running with scissors.
3. If you bring snacks, bring enough for everyone.
4. Zeroth rule of gun safety: Don’t let the government get your guns.
5. Rule one of gun safety: The gun is always loaded.
5a. If you actually want the gun to be loaded, like because you’re checking out a bump in the night, then it’s empty.
6. Rule two of gun safety: Never point the gun at anything you’re not willing to destroy.
7. Rule three: Keep your finger off the trigger until ready to fire.
8. Rule the fourth: Be sure of your target and what is behind it.

(Hmm a few extras seem to have crept in.)

Tranheuser Busch

Those asshats either get the world, all time record for obtuseness…or this is deliberate somehow. And their namby pamby non-apology that amounts to a “We’re sorry you’re too much of a knuckledragger to appreciate what we did” simply underscores the stupid. It’d be like a 400 pound golfer wearing spikes tromping on your instep and being utterly clueless why you are screaming in pain.

Me

I wrote that months ago, and they are casting about now trying to figure out how to get people to come back. However the one thing that might work (an abject admission that they were wrong to even think of doing this) is the one thing they cannot do. Because the decision makers are woke and cannot conceive that they were actually fuckheadedly wrong to do this.

Spot Prices.

Kitco Ask. Last week:

Gold $1,960.40
Silver $24.41
Platinum $945.00
Palladium $1,280.00
Rhodium $4,800.00

This week, markets closed as of 3PM MT.

Gold $1,943.60
Silver $23.73
Platinum $930.00
Palladium $1,290.00
Rhodium $4,750.00

Gold and silver still farting about in roughly the same places. Someone with a big futures presence can make some money off of these moves…some poor guy who bought physical metal a few months ago ain’t making squat (it has to rise enough to cover the buy/sell spread, which can be pretty wide). PGMs mixed.

More on Sines and Cosines

As you may have realized, I generally start to write a post by copying the one from five weeks earlier and modifying it…that way I can recycle my eagle pictures. (Right now that’s six weeks, because the Darwin Memorial Post doesn’t have an eagle on it.)

So I get to see an older science (or math) post of mine and I will sometimes read through it. When I did that this time I found multiple typos (not so bad) and also one glaring arithmetic error…that no one one called me on. (I’ve gone and fixed it.)

Makes me wonder.

Anyhow, we left off realizing that a complex number can be thought of as representing an angle, and that multiplying any complex number (including those that are pure real or imaginary) by another complex number, produces a complex number that represents the sum of the two angles. If one of the complex numbers has a magnitude of 1, the effect of the multiplication is to simply rotate the other number by the first number’s angle.

OK, those sentences were a bit complicated, so let’s show some examples: Take two complex numbers:

A = <a, α>
B = <b, β>

The first complex number A, is a units long at an angle of α radians, and similarly for B. (And a quick reminder. For clarity, I’m using angle brackets to indicate polar coordinates–a distance and an angle–but this is not any kind of standard notation.)

If you multiply those two complex numbers together, you get <ab, α+β>. The magnitudes get multiplied, the angles get added.

If b just happens to be 1, then you have an answer of <a, α+β>. In other words multiplying by B just means rotating A by the angle β.

And if both a and b are 1, you are simply adding the angles together to get another answer that has magnitude 1; it’s a pure rotation.

As I said before this is not just a geek parlor trick; it underlies much of electrical engineering.

Also discussed, a lot, last time were the sines and cosines of angles. I stated that some angles have obvious sines and cosines. For example, zero degrees measured on the unit circle has no y value, so its sine is clearly 0. And it’s x coordinate is 1, so the cosine is 1. Similarly 90° or π/2 radians is pure y coordinate with no x component, so its sine is 1 and its cosine is 0. You can easily figure out 180° and 270°, and of course 360° is the same as 0°.

But those aren’t the only obvious angles. 45° (or π/4 radians) is another obvious one. Why? Because you know the x and y coordinates of 45° on the unit circle are the same number, so it forms an isosceles right triangle…the two “legs” (the sine and cosine of 45°) are the same.

You can then get the rest of the way with Pythagoras. Squaring the two legs gives you the same number too, and the hypotenuse is 1:

1² = (sin(45°))² + (cos(45°))²

But the cosine and sine are equal so just replace cosine with sine (and 1² with 1):

1² = (sin(45°))² + (sin(45°))²
1 = 2(sin(45°))²

Dividing through by 2 we get:

½ = (sin(45°))²

Taking the square root of both sides:

√½ = sin(45°) = 1/√2

So we know that the sine of 45° or π/4 radians is one divided by the square root of 2.

But you do not want to divide by an irrational number if you can possibly avoid it, because they have an infinite number of digits, so think about writing that one out doing long division…you’re going to be multiplying long numbers a lot and subtracting them too.

So for historical reasons mathematicians never let a square root sign be in the denominator. You fix this by multiplying top and bottom by that number:

(√2 x 1) / (√2 x √2) = √2/2

Which works out to roughly .707.

So if you’ve done trig for any length of time when you see 45° (or 135°, or 225°, or 315°) you just automatically know the sine and cosine will be 0.707 (or -0.707).

You can use similar reasoning for 60°. That angle is the angle for an equilateral triangle, one where all three sides are the same length, and all three angles are 60°. So the origin (0,0), the (x,y) for 60 degrees on the unit circle, and (1,0) form an equilateral triangle. But the (x,y) are just (cos(60°), sin(60°)). The top angle of the triangle is half as far to the right as 1,0 is, so cos(60°) must be 1/2. You can then use Pythagoras on half of the equilateral triangle to show that the sine must be √3/2, or about .866.

(I must apologize for this diagram, it only shows half of the equilateral triangle. I found that any diagram that showed the whole triangle made the sides of length 2, and I thought that would be even more confusing than this diagram is.)

For 30° you have a mirror image of 60°, and you can swap the sine and cosine.

OK, what about the other angles?

Not every problem you’re going to want to solve with trig is going to involve one of these angles. In fact, if one ever actually does find oneself doing these angles, one is probably working a story problem!

It turns out that there are formulas for computing the sine (or cosine) of half of a given angle. So you could find the sine and cosine of 15° by running the formulas on 30°, and you could get the sine and cosine of 7.5° by running the formulas on that. You can get as small an angle as you like by repeatedly doing half angles. And then you can add those angles together in some combination with other formulae. For instance to get the sine and cosine of 11 1/4 degrees, run the addition formulae on the sines and cosines of 7.5° and 3.75°.

Actually you know how to do the addition formula: just turn the angles into complex numbers and multiply them together. The real part is the cosine of the sum of the angles, the imaginary part is the sine of that same sum.

Many people spent countless hours building up tables of sines and cosines this way, and back in those days they had to work longhand multiplying six digit numbers together just to get halfway decent results after rounding, and reusing the rounded numbers to do the next computation.

But there IS another way. It may not be easier but you can go directly to the sine and cosine of the angle you want instead of having to build off of other, known numbgers. So maybe it is easier because you could compute the sine and cosine of 11.25° directly instead of first computing the sine and cosine of 7.5° and 3.75°.

And I want to tell you what it is, and I was planning to do so tonight, but for some reason the Wordpus editor is slowing down on me more and more, and I’ve already restarted Firefox four times. It’s now slow enough that backspacing over one letter takes a second or two. And I have to wait for it to catch up as I type.

So I’ve run out of patience. Next time, the direct way to get a sine or a cosine. A hint: the sine and cosine are “infinitely differentiable.” (Not much of a hint, for people who haven’t taken calculus.)

Obligatory PSAs and Reminders

China is Lower than Whale Shit

Remember Hong Kong!!!

中国是个混蛋 !!!
Zhōngguò shì gè hùndàn !!!
China is asshoe !!!

China is in the White House

Since Wednesday, January 20 at Noon EST, the bought-and-paid for His Fraudulency Joseph Biden has been in the White House. It’s as good as having China in the Oval Office.

Joe Biden is Asshoe

China is in the White House, because Joe Biden is in the White House, and Joe Biden is identically equal to China. China is Asshoe. Therefore, Joe Biden is Asshoe.

But of course the much more important thing to realize:

Joe Biden Didn’t Win

乔*拜登没赢 !!!
Qiáo Bài dēng méi yíng !!!
Joe Biden didn’t win !!!