A question about sine and cosine Watch

j09
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I've just been wondering about this and I'm wondering if anyone could tell me...

If sine and cosine are calculated by an infinite series
http://upload.wikimedia.org/math/e/3...34c422a2a0.png



How is it that we can get a rational result when we find the sine and cosine of some numbers? (like pi, pi/2 etc.)

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Sine
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what about me ? :awesome:
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nuodai
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For a start \pi and \dfrac{\pi}{2} aren't rational numbers :p: It's just one of those wonderful things about maths.

Additionally, \displaystyle \pi = 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \cdots + \frac{4.(-1)^{r+1}}{2r-1} + \cdots

Which is awesome.
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SimonM
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Infinite series can sum to rational numbers?
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musabjilani
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Consider the geometric series with first term = 10 and common ratio = 0.5.

The sum to infinity = a/(1-r) = 20.

There, the sum of an infinite series equals a pretty much rational number.
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musabjilani
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(Original post by nuodai)
For a start \pi and \dfrac{\pi}{2} aren't rational numbers :p: It's just one of those wonderful things about maths.

Additionally, \displaystyle \pi = 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \cdots + \frac{4.(-1)^{r+1}}{2r-1} + \cdots

Which is awesome.
Maybe the OP meant sin(pi), sin(pi/2) etc, lol
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nuodai
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(Original post by musabjilani)
Maybe the OP meant sin(pi), sin(pi/2) etc, lol
I had an inkling that this might be the case... in which case the OP need look no further than a circle of radius 1 and origin (0, 0)... then it becomes more or less self-descriptive.
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j09
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(Original post by musabjilani)
Consider the geometric series with first term = 10 and common ratio = 0.5.

The sum to infinity = a/(1-r) = 20.

There, the sum of an infinite series equals a pretty much rational number.
But I don't think the series for sine and cosine have a common ratio...
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j09
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(Original post by nuodai)
I had an inkling that this might be the case... in which case the OP need look no further than a circle of radius 1 and origin (0, 0)... then it becomes more or less self-descriptive.
Yes that is what I meant, although I don't see how the way of displaying sine and cosine in a circle relates to calculating them as an infinite series :confused:
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Oh I Really Don't Care
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(Original post by j09)
Yes that is what I meant, although I don't see how the way of displaying sine and cosine in a circle relates to calculating them as an infinite series :confused:
It dosen't - they can be shown to be equivalent though.

(Original post by j09)
But I don't think the series for sine and cosine have a common ratio...
how can you accept that that sums to a rational number nut the series epansion for trig functions does not?
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