Is there an equivalent of cosx=1-(x^2/2) for the sin function

[Physgeek64]

Hi, i was just wondering since cosx=1-(x^2/2) is there a similar formatted formula for sinx??

much appreciated :) :)

 

[GFauxPas]

One of the definitions of cosine is:

$\cos x = 1 - \frac {x^2}{2!} + \frac {x^4}{4!} - \frac {x^6}{6!}+\ldots$

going on forever. If you take only a finite number of terms, then you'll have an approximation.

The corresponding series (infinite sum) for sine is:

$\sin x = x - \frac {x^3}{3!} + \frac {x^5}{5!} - \frac {x^7}{7!}+\ldots$

Google "factorial" if you haven't seen "n!" before.

 

[Physgeek64]

ahh okay! we're getting round to this in maths after had term. does it set up a GP??

so would it end up as x+(x^3/3) as an approximation?? xx

 

[.Scott]

No. Using only the first 2 terms, it would be: $x-{x^3\over6}$.

 

[Mentallic]

ahh okay! we're getting round to this in maths after had term. does it set up a GP??

Well no. Your first hint as to why it isn't a GP is that if it were, we would most definitely be applying the formula for the GP of an infinite sum. That would then mean that sin(x) could be easily represented as a simple fraction in terms of x and that would change everything in maths.

Your second hint is that if you divide the first by the second term, the second by the third, etc. you won't get the same result each time, so it can't be a GP.

so would it end up as x+(x^3/3) as an approximation?? xx

x-x³/3 :)

 

[Physgeek64]

Well no. Your first hint as to why it isn't a GP is that if it were, we would most definitely be applying the formula for the GP of an infinite sum. That would then mean that sin(x) could be easily represented as a simple fraction in terms of x and that would change everything in maths.

Your second hint is that if you divide the first by the second term, the second by the third, etc. you won't get the same result each time, so it can't be a GP.


x-x³/3 :)

okay. Thank you so much! I'm doing these physics papers and they're non-calculator and a lot of the involve using sin and cos for non-standard angles. Is there one for tan as well?? xx

 

[.Scott]

x-x³/3 :)

I'll take that smiley as an exclamation point!
$x-{x^3 \over 6}$

 

[.Scott]

Is there one for tan as well??

Aside from using sin/cos, there is this:
$tan(x)=x+{x^3\over 3}+{2x^5\over 15}+...$
See http://en.wikipedia.org/wiki/Taylor_series. About half way down there is a list of trig functions.

 

[Mentallic]

I'll take that smiley as an exclamation point!
$x-{x^3 \over 6}$

Yes that's exactly what I was aiming for haha

 

[Physgeek64]

Thank you! xx