How does a calculator find sinx? 

the most egregious programming tutorial ever

My favorite part is how it still uses a trig function

Funnily enough, I wasn’t too hurt when you used curly braces instead of a colon. It’s a common mistake, sometimes it gets hard switching between languages very often. I was hurt when you called that symbol a “hashtag”

you are the howtobasic of mathematics. lol

Finally, a channel does a better explanation of the Cordic algorithm than just "rotating the vector" to approximate a trig function, When rotations require trig functions. Brilliant video.

Hey there, awesome video, but I did just want to give a pointer. Using a variable called d next to x, y, or phi is generally considered an abuse of notation since it looks closer to an infinitesimal rather than actual variable.

This is all well and good. But to any future programmers watching this, please do not use weird Unicode math characters in your code, just use the names of things, like phi, theta, delta, using these symbols will drive anyone that isn’t a heavy math user mad when trying to read your code.

1) I think even if it's what algorithm is really used, there are some fine details about things regarding precision. Because if it shows 6 decimal places, then all of them should be correct. But error in cycle accumulates 2) Your code won't work for angles > 4pi 3) Question in the beginning was how do you calculate those without calculator. But then you pull out from somewhere: some constant which is limit of product (which is also you need to calculate without calculator), and table of 50 arctan, which you also need to calculate. I think more plausible way to calculate sin/cos without calculator to use angle halving formulas, and rotate by pi/2, pi/4, pi/8, pi/16 and so on.

If someone doesn't want to use pow function, the powers of 2 can be achieved by taking 1 and shifting it a few bits (remembering that 2^(-n) is the same as 1/2^(n))

Didn't understand this, will have to watch again later. But when it comes to approximating sin and cos, I find that this is a good plan: 1) Add or subtract multiples of 2*pi until you're in the range -pi to pi. 2) Map the angle to the first quadrant and remember what that will do to the sign of the final result. 3) If you're dong the sin or cos of an angle greater than pi/4, do the cos or sin of the complementary angle. With those three steps, we've guaranteed that our angle is no more than 0.785 radians. We can Taylor series it and get a good approximation within just a few terms. But we can take it even further: 4) Pre-calculate some sines and cosines of angles like pi/4, pi/8, etc. Save them as constants to whatever arbitrary degree of precision you like. 5) Remember your trig identities, like sin(a+b) = sina*cosb + coa*sinb, and cos(a+b) = cosa*cosb - sina*sinb. With those in mind, suppose you want to calculate sin(3*pi/16). Well, that's sin(pi/8 + pi/16), and if you've precalculated sin(pi/8), then you just have to calculate sin(pi/16) and cos(pi/16) and do the trig identities. And since pi/16 is a little under 0.2, the calculations for sin(pi/16) and cos(pi/16) will converge very quickly.

Excellent video mate! However, wouldnt a taylor series be easier for a calculator to deal with?

Instant subscription. Perfect intro into math amd coding combined for oeople unfamiliar with it. Keep it up!

this guys gonna be huge in the future

8:08 the ** operator also works. i.e: 2**(-n)

From the thumbnail, I thought it would be how modern calculators give symbolic answers for special cases when it recognizes them. IAC, what you described is called the CORDIC algorithm. It needs one iteration per bit of the answer, so 55 iterations seems right as that matches the mantissa of a double precision floating point value. CORDIC _can_ be implemented using only addition, subtraction, bit shifts, and table lookups -- no multiplication or division. Your code doesn't exploit this, and in fact uses division gratuitously. (division being horribly slow even on modern CPUs). This makes it the preferred algorithm for low-end calculators that use 8-bit microcontrollers. For a more capable CPU, the Taylor series takes fewer iterations and will need fewer as the angle is smaller.

For me, I would do either of the following: 1) Draw a right angle triangle with an angle of 1. 2) Use the Taylor Series.

Wow this vedio is very underrated. Excellent subscribed

Amazing video, I wish you the best your future efforts, and I can only hope you keep this quality up!

Damn, i didnt know howtobasic was a mathematician 💀

thanks a lot. this has been a question at the back of my mind for a lot of time

In degrees, use angle bisection as approximation. In radians, use the power series.

Pretty cool! Though if we don’t have functions for sine and cosine, shouldn’t we also not have functions for arctangent? Or is this actually the way computers calculate it?

or simply use binomial expansion of trig functions, define the function, replace the x with the variable name in the function parameter, keep writing as many terms as you can then you will get almost identical results to real values

Hello, what app do you use for that blackboard? I thought it looked very cool.

I remember in the 70's my dad brought home a TI calculator that had trig functions. Being about 8, I had no idea what they mean but I thought it was interesting that the calculator would take a couple of seconds to handle these functions. I made it my goal in life to be able to use all the functions on a calculator (it also had log as well.) But always wondered why it took so long to calculate sin, now I know.

Something worth noting here is, you still need to calculate arctan(2^-n) somehow, which is also a trig function. However, given this is very close to 2^-n, you can simply remove arctan for larger order terms, and perhaps hard-code first few terms to further decrease error.

You can just use tailor series, it works well with small numbers

I love the part where you used wolfram alpha to make you own trigonometric equation

Cool video! If you want to make those print statements a little easier to write and more readable, you can put an 'f' before the quotes and use curly brackets to avoid needing the str() functions. As in print(f"sin({θ}) = {y}")

7:54 Python uses the same double-asterisk operator as FORTRAN for exponentiation, so 2ⁿ would be written as `2 ** n`. Math.pow() always returns floating point numbers as a result, whereas the double-star operator will return integer values when appropriate.

this channel is a gem how i just saw this

What will you do? A B C or D? A: You can always go to the park B: You can always get to work on time C: You can always make a PERFECT triangle D: You go to Paris every year E: you ALWAYS get what you want

Imagine if a business major sees this. I think they’ll explode. Math majors may be sad, depressed, lonely, and overworked, but at least we can understand shit like this!

literallly just use taylor’s stratagey which is: sin(x) = x-(x³/3!)+(x⁵/5!)-(x⁷/7!)+(x⁹/9!), etc

Great video bruv! Just remember me when you have millions of subscribers😃

Acktually the sin is calculated using multiple techniques. Firstly, you only need to calculate the first quarent of the sin. Since other quarent can be calculate using trig. Secondly, look up table is used for common value like π/12, π/6, π/4, π/3, π/2 and more. Thirdly, values are close to 0 are return without calculation. Depend on how accurate the approximation need to be, cordic and Chebyshev polynomials can be use.

Iteration isn't the fastest method and there's a chance that change never reaches zero because of finite precision. It's better to use a polynomial or rational function. See Computer Approximations by J.F. Hart et al.

How do you get rid of the atan() function in the code? We're not supposed to use trig functions here, unless there is a video explaining how to approximate atan() without other trig functions!

Nice video. I didn’t understand everything but it was pretty interesting 👍

This means that the calculator needs to have a arctan(x) table in the memory or defined somehow for it to calculate sin(x)?

why can't we use infinite series expansion of sin, taking the first n terms such that it gives answer within accepted error limit?

I coded up CORDIC many years ago and was going to implement it in a FPGA but got bogged down with the floating point conversions.

being an actual python programmer, seeing the beginner tactics (like concatenation instead of functional strings or using Unicode characters as variables, or printing instead of returning) made me remind myself that beginners don't need to follow python conventions when their methods work. This was before I noticed you used curly brackets. (no hard feelings, great video)

i thought it used a parabolic approximation for 0-pi/2. then reflected and rotated that as necessary

The most underrated chanell on the platform

instead of math.pow, you can do 2**-n, no idea if it has the same time complexity tho

I do get some weird values. π/4 stops after 2 iterations, but ends up at the really wrong value (0.6072529350088812 instead of 0.7071067811865475). And cos(0) is really wrong, after 1 iteration. For π/2 the sin and cos are fine, but understandably the tan value is a bit wonky.

This was really a tutoriel that I watched with curiosity until the end. I liked both the math and computer part very much. My only question is, cos(arctan(1)).cos(arctan(2)).cos(arctan(3))... I think it is not appropriate to calculate it on the computer. Because we used trig again? Also i _think_ you can use Taylor Series of sinx , cosx, or tanx for example: sinx ~ x -x^3/3! + x^5/5! -x^7/7!

cool approximation of sin cos and tan, impressively interesting approach to programming it tho

4:51 I understand how the arctan values can be precomputed, but how do you calculate the cosine?

the math was pretty awesome, but I'm pretty sure using unicode characters, as theta is not a good programming practice. you should stick as long as you can to ASCII to name variables and functions

If you're gonna use a trig function anyway (atan), why not just def trig(theta): return math.sin(theta)

8:03 that's not "to the power of", that's "xor". XOR is a weird binary thingy, if you want "to the power of", use ** instead of ^

peak cinema of math

Very nice!

Fortunate that people were able to use wolfram alpha back in the day, despite not having a calculator

The whole point of the original CORDIC (published by Jack Volder in 1957ish) was to replace computationally heavy/expensive multiplication and division in old memory-poor computers with additions/subtractions and some table lookups. Logs were also possible. Though based on some obscure 17th Century mathematics it was still a damn impressive algorithm. The code here would not have worked efficiently on early computers and calculators. In fact, it would have defeated the whole point of the original CORDIC. Interesting, though.

Why does the algorithm for finding trig functions need you calculate arctan? How does it do that?

calculators actually have tables with all the sin values with the maximum precision they need. they dont directly calculate sin() because of perfomance

Confirmed, Wolfram Alpha existed before calculators did.

Better to pronounce the sign() function as SIGNUM. Not “sine” - because that would confuse it with sin().

"But wait, that requires cos and sin." "Aaaarerggghg!!!!!!!!!" Got me dying. 💀 Eggs in a blender.

11:00 this jump scared me slightly

this is amazing

Good video!

I’ve wanted to know this for a while

Video: How to make a trig function 8:45 : Ok first you have to use a trig function

And I thought for half a century that mathematicians and programmers have zero emotions ...

I always thought it used the Taylor series expansion

i love this

I noticed the brackets immediately and was very confused by it, I was like: Why didn't we just do this in c or somethin and why did he do that?

How does the calculator display it in a form like √2 /2 or 3π/2?

When I saw the brackets I died

Legend says the CORDIC isn't used anymore.

0:15 A

can't you just use maclaurin's expansion for the first couple terms

Imagine being in 1956 without a calculator and having a Python interpreter...xD

now make an infinite precision pi calculator

8:15 or you can use the ** operator

I had to stop watching a few minutes in because I couldn't focus afraid of a scene with wasted eggs and phone books sudddenly appearing.

imagine not waiting until deltamath was invented

Why are you so fucking funny lmao

Interesting video but I don't like the fact that this algorithm uses a trig function to define other trig functions. I think it's sexier to derive trig functions from lower level math abstractions.

I really liked the video but the python part made we want to bang my head

Why not use Taylor series approximations?

the frustrated AUURRGHHH 🥚

How does a calculate find atan(2^-n)?

Why use this approach over a Taylor / Maclaurin series?

I have subscribed

I thought calculators use Taylor's series. What's wrong with that?

How to find sin of whatever? Use tan. But how do I find the tan of whatever?

ok but how to calculate the atan then?

Okay, umm, how do you calculate arctan? You've kinda kicked the can down the road by relying on another function. Obviously you could do numerical integration but that would be slow as balls

I actually asked myself 2 days or so ago

But you had to use Wolfram Alpha to kind k...

why do we need k? can't we just do y/sqrt(x^2+y^2) in the end?

ERORR: division by zero line 7 and 13

Also me, who knows what sin 60 degrees is and also knows that 60 degree = 1.047 radian. so i just approx sin of 1 radian as sin of 60 degrees which gives me 0.86. I call that good enough and move on. ᕙ(⇀‸↼‶)ᕗ + 1 sub

Good vid. Have no clue what’s going on.

8:13 Could've just used the ** (double-star) operator. if you're worried about any performance issues.... IDGAF HE'S PROGRAMMING IN PYTHON FOR FUCK'S SAKE