AFAIC, $\sin/\cos$ functions take an angle and spit out the ratio of the opposite to the hypotenuse, in the triangle, written into a unit circle; because we can measure the angle in radians, we can plot these ratios on a graph, taking the $x$ axis to be the radians and the $y$ axis to be the radius of the unit circle, which is $1$. It is done easily enough for common ratios, like $\frac{1}{2}$ or $\frac{1}{\sqrt2}$ but how do we calculate the ratio, i.e. $\sin/\cos$, of any angle? I've heard about using Taylor series but I've not seen a proper explanation anywhere that would be coming from someone who actually understands it and knows about the limitations of the methods they're talking about.
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https://math.stackexchange.com/questions/501660/is-there-a-way-to-get-trig-functions-without-a-calculator, https://math.stackexchange.com/questions/395600/how-does-a-calculator-calculate-the-sine-cosine-tangent-using-just-a-number – Martin R Aug 06 '20 at 20:40
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1I'm not quite sure what you are asking. That ratio is the tangent of the angle. It's no harder to calculate than the sine or the cosine. You could [edit] the question to clarify just what you want. Then maybe we could help. – Ethan Bolker Aug 06 '20 at 20:42
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The geometric definition is hardly used, fwiw. Generally you would define $\sin$ and $\cos$ by their power series and then define $\tan$ from there without ever talking about triangles or ratios of sides. – George C Aug 06 '20 at 20:45
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Thanks @MartinR, that's what I was looking for! – matzar Aug 06 '20 at 21:16
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Thanks @GeorgeCoote what you've written was really helpful! – matzar Aug 06 '20 at 21:17