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If the $\cos$ function is based off of the ratio of the adjacent side of Euclidean, right triangle, with fixed hypotenuse length (such as the unit circle), then how does this correspond to a defined value when $\theta = \pi/2$ ? To me the limit as $\theta \rightarrow 90$ appears to be undefined (specifically 1/0).

Arguments against my view point: 1.) Once you hit 90 for a triangle to exist a new, right triangle will be formed, that is if the hypotenuse is a real number. Counterpoint: That's not the same angle, $\theta$, we used for our original function and when you hit 90 there's not going to be an adjacent side length of zero.

Nebo Alex
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Zduff
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  • This is a pretty good question. I would say that you are correct about there being a problem (you don't have a triangle), however you can extend it continuously to $\frac{\pi}{2}$. – Cameron Williams Mar 10 '16 at 02:44
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    The triangle definition of the trig functions are only valid for $\theta$ in $(0,\pi/2)$. Luckily that's not the only definition we have. –  Mar 10 '16 at 02:44
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    The sine and cosine functions can be defined in terms of projections of a hypotenuse (or radius or other fixed length) onto the $ \ x- $ and $ \ y-$ axes, which are never undefined. – colormegone Mar 10 '16 at 02:45
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    @RecklessReckoner I think they just mixed up which order the ratio goes in. – Cameron Williams Mar 10 '16 at 02:46
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    This answer may be helpful. – Blue Mar 10 '16 at 02:47
  • $\cos$ is defined rigorously using power series. – Henricus V. Mar 10 '16 at 03:47
  • Like Cameron said, the limit is $0/1$, not $1/0$. – Hans Lundmark Mar 10 '16 at 05:59
  • See this answer for a representation of a perpendicular right triangle. http://math.stackexchange.com/a/1678623/140156 – John Joy Mar 10 '16 at 17:17
  • Yeah the limit is 0/1. But the cosine function is continuous and does not exist at $\pi/2$ (in sole terms of triangles ratios). It's strange that this function is defined by it's power series like Henry said, because didn't these functions and their cartesian graphs exist before power series existed? – Zduff Mar 11 '16 at 17:52

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