problem: sin(10^40)

hamzah · March 21, 2004

how would I go about finding sin(10^40) ? [10^40 in radians]

Crash · March 21, 2004

Set your calculator in radians instead of degrees

Dave · March 21, 2004

I've been looking at the problem, and I can't really see a solution of the top of my head apart from just plugging it into Matlab/Mathematica. I've got a hunch that you could use the fact that 1/10^40 = 10^(-40) is such a small angle that you can use small angle approximation, but I'm probably totally wrong.

wolfson · March 21, 2004

Find 10^40 mod 2pi, since sin(2pi+x) = sin x.

hamzah · March 21, 2004

I did try sin[10^40 - int((10^40)/(2pi))]...but none of the calculators I own are accurate enough.

wolfson · March 21, 2004

You should get:

-0.569633400953636327308034181573569 or near to

Dave · March 21, 2004

from matlab:

>> format long

>> sin(10^40)

ans =

0.64678458842683

which isn't what you got. Don't know who's right though

wolfson · March 22, 2004

Imagine a graph of sin(x). It's positive for 2pi radians, then negative for the next 2pi radians, then positive, etc.

When you go along the x axis to the point 10^40, the graph must just turn out to be below the axis there. The angle (x) isn't negative, but the sin(x) value is; like when you have sin 210 degrees = -0.5.

Kedas · March 22, 2004

Sin(10^40 - 2pi*10^39)

So pi has to be at least 40 digits accurate to have some result in the right direction.

A calculator only has pi up to 15 digits? so can't be done with a normal calculator.

maybe with a mathematical trick you can solve it on a normal calculator.

wolfson · March 22, 2004

"So pi has to be at least 40 digits accurate to have some result in the right direction."

He was not after accuracy he was after a method of solution.

Thus:

Find 10^40 mod 2pi, since sin(2pi+x) = sin x.

Thus:

-0.569633400953636327308034181573569 +- (1e-12)

To compute use windows scientific calculator

Kedas · March 22, 2004

windows calculator (scientific mode) 'only' has pi up to 32 digit so you need an other method/program.

this accuracy is part of the solution method.

Dave · March 23, 2004

wolfson said in post # :
Imagine a graph of sin(x). It's positive for 2pi radians, then negative for the next 2pi radians, then positive, etc.

I think you mean every pi radians there, not 2pi.

windows calculator (scientific mode) 'only' has pi up to 32 digit so you need an other method/program.

this accuracy is part of the solution method.

Indeed. I suspect that this isn't the most efficient method you can use. I think the correct answer is the one I quoted above, since I've now verified it with matlab and mathematica - both of which are very capable of producing numbers to arbritary precision.

wolfson · March 23, 2004

When I said 2pi, I should have said pi. I seem to be making a lot of mistakes this evening! But i still say my negative is correct.

AntiMagicMan · April 21, 2004

I used Maple 9 and used evalf(sin(10^40)); and got -.5696334010 so i'm agreeing with Wolfson here. Mathematica 5 also gives -0.56963340.

Dave · April 22, 2004

sigh.

Even when I type it into a calculator, I seem to get the answer wrong - not once, but twice. It's all great - seems to be the story of my life

AntiMagicMan · April 22, 2004

I think most handheld calculators use taylor series approximation, there simply will not be enough terms for the sum to converge with a number that big.

Dave · April 22, 2004

Yes, I know. I wouldn't really expect a normal handheld calculator (apart from perhaps my TI-89) to be able to work it out.

My point is, I said above:

from matlab:

>> format long

>> sin(10^40)

ans =

0.64678458842683

and then proceeded to type it into mathematica and get the same answer (somehow).

AntiMagicMan · April 22, 2004

In[6]:=
N[sin[10^40], 50]

Out[6]=
-0.56963340095363632730803418157356872313292131914787

Dave · April 22, 2004

*shrug*

Don't know to be quite honest

AntiMagicMan · April 22, 2004

That is quite confusing, the standard input for maple and mathematica is radians... so I'm not really sure where the difference comes from.

Dave · April 22, 2004

I might've somehow set it to degrees, but I can't remember. Nevermind, it's hardly like it's important.

Kedas · October 28, 2004

For those interested: I kept the calculation accuracy of it high but added a pi that wasn't that accurate you can see that only at accuracy 41 we are getting close/stable.

with maple 8:

you get pi followed with the result of the calculation.

sin(10^40-2*p*10^39) with p=Pi > p:=evalf(Pi,70) accuracy=70

> p:=evalf(Pi,70);

> evalf(sin(10^40-2*evalf(p,70)*10^39),70);

p:=3.141592653589793238462643383279502884197169399375105820974944592307816

-0.5696334009536363273080341815742365755028674749735871234233590307534688

> p:=evalf(Pi,50);