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Yes, 355÷113 is very close to π, but is not equal to π
So how does it get things 'right' for cos(355÷113), but not right for sin(355÷113)?
And why is the error of π-355÷113 exactly the same as the error of sin(355÷113)?
I sense some fuckiness of how they handle π...
Also keep in mind that your calculator has limits. Real π is irrational but in a digital calculator it must be stored as a binary value in a chip, so it can't have infinite precision, it's not really π but a good-enough-for-most-uses approximation. The level of precision may vary from one calculator to another.
You can figure this out if you do π - 3, then multiply the answer by 10, subtract 1, multiply by 10, and so on subtracting the whole number value and moving the decimal point until you run out of digits. On my TI-36X π only has 12 digits: 3.14159265359
From memory..
3.14159265358979323846264233832795028841971693993751
There isn't really an issue here. The reason the cosine value is rounded to -1 while the sine value isn't rounded to 0 is because the cosine value is much closer to -1 than the sine value is to 0. The unrounded (or less rounded) values are cos(355/113) = -0.99999999999996441843 and sin(355/113) = -0.00000026676418906242. So while the sine value is about 10^-7 from 0, the cosine value is about 10^-13 from -1, 6 orders of magnitude closer. Your calculator's threshold for rounding is just somewhere between those magnitudes.
As for why the latter two calculations give identical answers, that's just a feature of sine itself: For very small inputs it's an excellent approximation of the identity function, f(x) = x. If you give it any input of similar size to π - 355/113, it'll more likely than not give you the exact same value back out. As x → 0, sin(x) → x. Try it out with other values like 0.0000000123456789.
Interesting. My TI-36X Solar version 3 from 1994 seems to get ~~gets~~ these calculations exactly right, within the 10 digit display, plus 2 hidden digits for extra accuracy.
Yes, my calculator from 1994 ~~is~~ seemed more accurate than this crap manufactured in 2009. Advertised to be of the same 10+2 digits of accuracy no less.
Edit: Sigh, reminders of over 25 years ago when I compared engineering rulers together and none of them perfectly matched up.
To be clear, when you say "exactly right", do you mean -1 and 0 or -1 and -2.6776418E-07? Because -2.6776418E-07 is the more accurate answer here. 10+2 digits of accuracy does round the cosine to -1 because its first 13 digits after the decimal are all 9s, while 10+2 decimals of accuracy for the sine should be -0.000000266764 (12 digits) rounded to -0.0000002668 (10 digits rounded), then displayed as -2.668E-07 - so you actually end up with some bonus accuracy in this case. Though that last 8 should round up to a 9.
I found the results of cos(355/113) to be odd, as I'd expect it to give me a value of 0.99849715, not 1.
Which begs a different question, since when does cos(π) = -1?
My Linux Mint 20 system wouldn't crank out a 1 until I simplified the calculator down to cos(22/7), and that was a positive one, not negative. Which matches the sign of my TI-36X, positive one.
Now we're getting down to some real brass tacks here, what's the chirality of trig functions, is cos(π) supposed to equal positive or negative one?
Edit: within whatever margin of error, QBasic gives me -1, Casio Calculator gives me -1, Linux Calculator gives me +1, and Texas Instruments gives me +1...
WTF?
That comes down to the calculator using radians while you're expecting degrees: cos(0°) = 1, and cos(355/113 degrees) = 0.99849714986386383364. The default for most calculators is to do trig functions in radians, and there we have cos(0) = 1 and cos(π) = -1. π degrees is much closer to 0° than 180° (which is equivalent to π radians), hence the answer for that being almost 1.
The OPTN button near the SHIFT button will probably let you swap between RADians and DEGrees
Oh hell, alrighty then! Sure enough my old QBasic 3D graphics engine from 1998 was working in radians!
Holy hell I forgot about that, it's been about 27 years since I actually looked at my first rendering engine!
I'll see myself out now, but thank you for the refresher! 👍
I almost guarantee you all my calculators are configured for degrees for these tests. I edited my comment after testing 4 different calculators.
Now I just don't know why two of them give me -1 and the other two give me +1.
Edit: I may have to learn more about this Casio, but still I'm getting conflicting results between other calculators, including modern Linux Calculator.
The correct answer in degrees is cos(pi) = 0.99849714986386383364. The correct answer in radians is cos(pi) = -1 (exactly). Any calculator giving you cos(pi) = -1 is definitely in radians mode - and if you mean you're getting cos(pi) = exactly 1, and not 0.998, then that should never happen in any mode, unless it just has two digits of accuracy. Which I doubt any calculator with a 'cos' button has ever had.
For the record, if using sine, you should have sin(pi) = 0.05480366514878953089 if in degrees mode, or sin(pi) = 0 (exactly) if in radians mode.
Note, I edited my comment. Now I'm reminded of when I tested engineering grade rulers against each other like 25 years ago and nothing perfectly matched.. ☹️