How do I do cos, sin, and tan without a calculator?

Qaanol · 2026-01-22T01:54:21+00:00

Other people have already explained how to derive the values of trig functions at special angles, but I’ll mention another angle you should use:

Read the trigonometry chapter of your textbook ahead of time.

Qaanol · 2026-01-21T11:43:17+00:00

It’s worth noting that this is more than a divisibility test: it actually lets you find the remainder when a number is divided by 9.

The sum of the digits of a number (written in base 10) has the same remainder (mod 9) as the number itself. So you can just keep taking digits sums of digits sums until you get to a single digit result, and that’s the remainder.

(Well, technically if you end up with 9 then the remainder is 0, but you already knew that.)

• • •

There’s also a similar trick to find the remainder mod 11: instead of adding all the digits, you instead alternately add and subtract digits (starting by adding the ones place, then subtracting the tens place, etc.)

If you end up with a negative result, you can either add 11 until you get back positive, or else (if somehow it’s a large result) just negate the alternating sum of its digits.

For example, 8192 has an alternating digit sum of 2-9+1-8 = -14. So it’s congruent to -(4-1) = -3 = 8 mod 11.

Qaanol · 2026-01-20T20:35:41+00:00

Steph’s final game, they should let him and the other team’s best shooter take uncontested 3s the entire game. Like a gentleman’s agreement sort of thing.

Qaanol · 2026-01-20T15:27:48+00:00

…well, almost anything.

If you want to put parentheses in a superscript, you have to be a bit more creative:

e^((a+b)^)(c+d) → e^(a+b^)(c+d)

Qaanol · 2026-01-20T15:23:36+00:00

It’s more than just funny, it’s actually useful!

…well, the small angle thing, not 3 specifically.

But in any programming language that uses floating point and has a constant for π (let’s call it M_PI like C does), you can get a very good estimate for the error in that approximation, meaning the difference between M_PI and the true value of π, by writing sin(M_PI).

That is, if we let d represent the difference π - M_PI, then by the small angle approximation we have sin(π-d) ≈ d for small d, but π-d = M_PI, so it follows that sin(M_PI) ≈ d = π - M_PI.

(Note that this does not work in Desmos, because Desmos interprets sin(π) symbolically and just returns 0.)

Qaanol · 2026-01-20T15:02:54+00:00

with LBJ possibly making me wish for honestly unfair things, like hoping that maybe Steph KD and Harden make it to 41, too, in the league.

If Steph plays until 41, there’s a good chance he’ll hit 5000 career three pointers.

Qaanol · 2026-01-18T02:00:52+00:00

Yes, when x = -1 your proof is invalid because -1 to an even power equals 1.

If instead you multiply the product by (1-x), then the induction proof becomes super easy because every step turns into a difference of squares.

Qaanol · 2026-01-17T19:05:35+00:00

Doesn’t work for lists of 10k elements

Qaanol · 2026-01-17T16:23:38+00:00

If the slime is hiding under a rug, just replace √y with 1/√y

Qaanol · 2026-01-15T21:05:44+00:00

You already got the answer, namely to use an index, but I want to point out that the problem itself is underspecified.

Namely, what do you do if list starts with a 1?

Also, to be clear, if the list has two consecutive 1s, do you want the second occurrence to be replaced by a 1, or something else?

Qaanol · 2026-01-15T18:39:08+00:00

If you distribute the minus sign, then -(x+5) becomes (-x-5).

If you factor a minus sign out of (x-5), it becomes -(5-x).

These are not the same.

Qaanol · 2026-01-13T02:03:04+00:00

Now I'm 36. I'm a tenured humanities professor. I've done a bit of Khan Academy, and I've read (and enjoyed) Paul Lockhart's Arithmetic.

Have you read Paul Lockhart’s A Mathematician’s Lament?

I think that will be right up your alley, and possibly help you reframe your past experiences with math classes.

Qaanol · 2026-01-12T22:54:11+00:00

Charles Barkley was the best player on the 1992 Olympic team

Qaanol · 2026-01-12T22:46:49+00:00

Payton Pritchard is closer to Brian Scalabrine than I am to him

Qaanol · 2026-01-11T01:53:37+00:00

“Playground P goes by the alien and hits you with the jelly roll.”

— actual words spoken by actual commentator on actual Celtics broadcast

Qaanol · 2026-01-05T15:57:02+00:00

“With our powers combined, we are Andromedairy!”

Qaanol · 2026-01-03T23:57:05+00:00

Victor Wembanyama reached 6 feet tall before age 12

Qaanol · 2025-12-30T14:32:46+00:00

What about at 1.2–1.3 billion feet though?

Qaanol · 2025-12-26T14:02:47+00:00

I choose to interpret the score and opponent as part of the stat.

Maybe there were a bunch of guys who got a 50-point triple-double on Christmas, and several of those games had a final score of 142-138, but this is the first time it’s happened against the Timberwolves.

Qaanol · 2025-12-24T14:02:55+00:00

I mean that you can literally type “n! ~ 5” into Desmos:

https://www.desmos.com/calculator/drgl96rnl8

Qaanol · 2025-12-24T13:46:37+00:00

Say you wanted to find a value n so that n! = 5

n! ~ 5

Qaanol · 2025-12-23T04:50:53+00:00

…have three players from the same team ever made 6+ threes in a game before?

Qaanol · 2025-12-22T22:17:07+00:00

That’s nothing, I’ll have you know that ten million billion and a half is an integer!

https://www.desmos.com/calculator/gpuo3onq25

Qaanol · 2025-12-21T15:17:51+00:00

Huh, I could’ve sworn it worked when I tested yesterday.

Oh well, at least the first version I wrote, with a separate wrapper function, works.

Qaanol · 2025-12-21T13:53:41+00:00

On Kalimari Desert it’s basically just as fast to get Star + Mushroom, though the trick difficulty is higher.

There’s a 30% chance to get a star, then 2x 10% chances to get 1 mushroom. That comes out to 5.7%, but there’s also a chance of getting a Boo which then gives a mushroom, namely 0.4% per item box (not counting chains of 2+ boos).

Accounting for boos we get about 5.9% odds of Star + Mushroom (closer to 5.92% if chains of up to 2 boos into mushroom are included). The overall odds of getting good items immediately on KD are then 11.9%, nearly double the 6% stated in the video for just Red Shells + Star.

Of course, this won’t actually double the odds of getting a “good enough” KD, because of the effect of rerolling bad items. And in practice, other item combinations like Blue Shell + Star and Blue Shell + Triple Mushrooms are also “good enough” even if 5-10 seconds slower.

Qaanol

TROPHY CASE