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I am reviewing basic trigonometry and came across a nice example that uses an equilateral triangle (sides of length r), cut in half vertically, to demonstrate that cos(60) = r/x = r/(r/2) = 1/2. From high school, I had always thought that the unit circle was somehow more significant for deriving trigonometric values, but now I find myself unsure, realizing that the trig functions are simply representations of ratios of sides of triangles of any size. Why is the unit circle the preferred choice to derive values over another circle of radius (hypotenuse) r?

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    $\begingroup$ Because when you make the ratio, the denominator is $1$. That's a friendly ratio. $\endgroup$ – Jean-Claude Arbaut Sep 20 '14 at 22:33
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Because the "unit" (whatever) is the preferred (whatever) to work with.

In linear algebra, you can work with orthogonal vectors, but it is easier to work with "orthonormal" vectors, that is unit orthogonal vectors that have been divided by their norms.

And as Thomas Andrews pointed out, you can work with circles, but it's easier to work with (unit) circles whose size have been divided by their radius, and therefore have a "radius" of one.

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  • $\begingroup$ I think this makes sense. Because a trig function is a ratio, and a ratio has a max value of 1, suggests that when visualizing the points of the functions one can and should represent them with r=1, even if it does not provide any advantage in terms of calculating actual values of the functions. $\endgroup$ – jwalk Sep 20 '14 at 23:19
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Fundamentally, it is so much easier using the unit circle.

But there is something deep going on. The complex number formula $e^{ix}=\cos x + i\sin x$ traces the unit circle, and the fact that the path traced by $e^{ix}$ for $a\leq x\leq b$ is of length $b-a$ are deep reasons.

But you can define them with any circle. You just always end up dividing by the radius in those cases.

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  • $\begingroup$ Meaning, it is easier to visualize the values, and not actually when calculating them? Specifically, I find using an arbitrary value r in the case of cos(60) more meaningful, though this is a trivial example. $\endgroup$ – jwalk Sep 20 '14 at 23:30
  • $\begingroup$ Meaningful is a tricky thing. The reason it doesn't matter what radius we use is due to the similarities of triangles. As I pointed out, there are advanced ways in which $\sin x$ and $\cos x$ are "about" the unit circle. More basically, the equation $\cos^2 x+\sin^2 x = 1$ is a statement that $(\cos x,\sin x)$ is on the unit circle. That seems pretty "meaningful" to me, but it is a subjective standard. $\endgroup$ – Thomas Andrews Sep 20 '14 at 23:36

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