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I'm trying to wrap my head around sine, cosine, and tangent. I'm aware that they're commonly defined in high schools as ratios of the various parts of triangles set in the unit circle, but that's not particularly intuitive.

Can the trigonometric functions be expressed, explained, or proven in terms of something more intuitive, say, arithmetic? Is there a way to work out sine, cosine, and tangent on paper, without the aid of a calculator or computer?

How were they first proven or discovered? What proof might have been worked through to reach the trigonometry we have today?

Edit:

I discussed this question with a friend today. I think ultimately, what I'm looking for, is a way to work out the sine, cosine, or tangent of a given angle using paper and pencil.

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    The trig functions were certainly first discovered through geometry, which many people consider to be at least as intuitive as arithmetic.2012-08-23
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    You may be interested in the following [Wikipedia article.](http://en.wikipedia.org/wiki/History_of_trigonometry) Originally, a circle of fixed radius was chosen ($60$, $10000000$, more exotic choices) and people were interested in the length of the chord, given the length of the arc. So originally sine was a length, not a ratio.2012-08-23
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    @GerryMyerson - Of course, updated question.2012-08-23
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    The high school definition you mentioned is pretty geometric, isn't it?2012-08-23
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    What about the triangle definition do you not consider geometry?2012-08-23
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    Eh, true, it's late and I'm tired. My main goal is to figure out how to evaluate sine of X on paper.2012-08-23
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    There are formulas that approximate trigonometric functions with arbitrary given precision, that have been used and perfected since the ancient times. Those include infinite summations, infinite products, and such. It sounds like you might call them arithmetic approach to trigonometric.2012-08-23
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    @ThomasAndrews - Forget geometry for a sec. See my edits.2012-08-23
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    You need a lot of time to calculate trig functions with a generic argument on paper, but it is not impossible. For rough calculations, you just remember the function values at some special angles ($\frac\pi6$, $\frac\pi4$, $\frac{2\pi}3$ etc.), and then use formulas, e.g, for $\cos(x+y)$, $\sin\frac\alpha2$ etc.2012-08-23
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    $x-\frac{x^3}{6}$ is a good (and reasonably quick to compute) approximation then. Just convert all angles to their acute equivalents and go crazy. In general this has to do with what's known as the Taylor Series of $\sin(x)$, but you won't get to that till calculus.2012-08-23
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    Another easy way to approximate the sine for small arguments: $$\sin\,x\approx x\frac{60-7x^2}{60+3x^2}$$ For cosine, $$\cos\,x\approx \frac{72}{x^2+12}-5$$ For tangent: $$\tan\,x\approx x\frac{15-x^2}{15-6 x^2}$$ From that point, you can use the double-angle identities repeatedly for evaluating sines and cosines and tangents of larger angles.2012-08-23
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    @Moshe: the answer to your edited question is that it depends on what kind of angle you're given and what kind of answer you want (an exact answer? A good decimal approximation?).2012-08-24
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    Have you learned about Taylor/McLaurin series yet? That will give you the understanding of why it cannot be a finite simple formula.2012-10-17

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