Approximate tan inverse (0.1) to three decimal place accuracy using the Maclaurin series tan inverse x

- blackstreet23

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- blackstreet23

@PlasmaFuzer

- blackstreet23

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- blackstreet23

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- blackstreet23

@Shalante

- blackstreet23

##### 1 Attachment

- blackstreet23

I did this so far
The book says that the error to 3 decimal places is
\[(0.1)- \frac{ 0.1^3 }{ 3 }\]

- blackstreet23

however when i put (0.1^3)/3 in my calculator the answer is 3.3333 * 10^-4
that is less that 0.5 * 10^-3

- blackstreet23

so why could that be the answer? wouldn't it just be 0.1 ?

- blackstreet23

@Hero

- IrishBoy123

the question asks you to do it using the maclaurin for cos x
is that what you did?

- blackstreet23

yes

- IrishBoy123

|dw:1442179486992:dw|
terrible graphics i know but it seems you plugged your numbers into the expansion for\(\tan^{-1}x\).

- freckles

I wonder why it says to approximate arctan(0.1) using the maclaurin series for cos(x)
and here it says use maclaurin series for arctan(x)
I think we would want to use the series for arctan(x) since we are apporximating arctan( something)

- blackstreet23

yeah i did.

- freckles

but it really does say cos(x)?

- blackstreet23

tan inverse (0.1)

- freckles

You know I'm trying to tell you that there is conflicting information here right?

- blackstreet23

ohhh I see

- blackstreet23

my bad the one i copied was wrong

- blackstreet23

but the question here is right

- freckles

ok this question makes more sense to me :)
Approximate tan inverse (0.1) to three decimal place accuracy using the Maclaurin series tan inverse x
\[0.1-\frac{0.1^3}{3} \text{ is a pretty good approximation for } \arctan(0.1) \\ 0.1-\frac{0.1^3}{3} \approx 0.099\overline{6} \\ \text{ while } \arctan(0.1) \text{ to 6 decimal spots is } 0.09966\]

- blackstreet23

Why is the the answer they have on the back of the book is
(0.1)−(0.1^3/3)

- blackstreet23

but the way I learned is by taking each monomial of the series and substituting the value of x

- blackstreet23

and (0.1^3)/3 is less that 5*10^-4

- freckles

yep but then you have to add the terms in the series

- freckles

where does 5*10^(-4) come from ?

- blackstreet23

from a rule that says that the decimal places they are asking you for is equal to
0.5*10^-n
where n is the number of decimal places

- blackstreet23

so i just simplified to 5*10^-4

- blackstreet23

and i need to find a term that is smaller than 5*10^-4 in the series

- blackstreet23

@PlasmaFuzer

- freckles

oh I don't remember that
I would have honestly found arctan(0.1) first which is approximately 0.0996
from the calculator then added the terms in the series until I got with in digits of that number
\[\arctan(x)=\sum_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{2k+1} \\ \arctan(x)=(-1)^0 \frac{x^{2(0)+1}}{2(0)+1}+(-1)^{1}\frac{x^{2(1)+1}}{2(1)+1}+(-1)^{2}\frac{x^{2(2)+1}}{2(2)+1} \\ + \cdots +(-1)^k \frac{x^{2k+1}}{2k+1}+ \cdots \\ \text{ so let's see if one term is enough } \\ \arctan(x) \approx x \\ \arctan(0.1) \approx 0.1 \text{ this is a really bad approximation } \\ \text{ how about two terms } \\ \arctan(x) \approx x- \frac{x^3}{3}\]
\[\arctan(0.1) \approx 0.1-\frac{0.1^3}{3} =.099\overline{6} \text{ this is an awesome approximation }\]

- freckles

the approximation only gets better if we add more terms onto that

- IrishBoy123

|dw:1442182723650:dw|

- anonymous

@blackstreet23 Hello... Freckles and irishboy have it right here... That formula you have listed (0.5*10^-n)I dont recall (though I think I can guess where it comes from), but if your issue is you want more accuracy simply add one more term to the approximating series no?

- blackstreet23

well i guess i will ask this one to my professor. Thanks guys for your help!

- anonymous

Sorry I couldn't help Im not sure what you are hung up on as far as the problem is concerned

- blackstreet23

ohh no problem! i know you did your best :)

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