if we take a look at value of , using a calculator we can say it is 89.427.
then if we go to , we get 89.9427
for it is 89.99427
is 89.999427.
So we have a pattern here using which we can predict the value of
for every zero after 100 , add a 9 after the decimal and let the 427 term remain same.
If anyone has an explanation please do share.
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This is not hard to explain. First, the function you're trying to compute is actually π180tan−1(x), where the π180 is just the factor that converts from radians to degrees.
First note that for ∣x∣<1 we have π180tan−1(1/x)=90−π180tan−1(x)=90−π180(x−3x3+5x5−⋯)=90−π180x+π1803x3−π1805x5+⋯ and now we're plugging in x=1/10n. Since x3,x5,… are tiny compared to x, we can get a very good approximation by cutting off those terms. So we get π180tan−1(10n)=90−π18010n1+π1803⋅103n1−⋯≈90−π18010n1≈90−10n57.3 and that's that. So: π180tan−1(100)π180tan−1(1000)π180tan−1(10000)≈90−10057.3=89.427≈90−100057.3=89.9427≈90−1000057.3=89.99427 and so on.
Hi! Just a couple of thoughts:
First, the function tan−1(x) approaches 90∘ as we input larger and larger numbers, so it makes sense that your values also get closer and closer to 90∘ .
I actually tried a similar procedure with an initial input of 200. I kept increasing this input by factors of 10 and saw I similar pattern: just adding nines at the front of the decimal part. I suppose this would happen with most large numbers.
We can't exactly say that adding nines is all we have to do however, because the numbers you listed have been rounded. The remaining digits do change with each calculation.
I'm actually not quite sure though what causes this "add a nine" behaviour. Not sure if it is an actual mathematical property or just the result of using a calculator. Hopefully someone else knows!
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Hi David, Thanks for your info about the fact that it doesnt need to be factors of 10 to show this pattern and I know that I have taken an approximation but still it is mind boggling that we can begin predicting values of tan x where it starts getting unpredictable.
Wow man What an observation, I cant even comprehend this
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Thanks