2017年3月4日 · In Calculus by Michael Spivak, the area of the unit circle is explored using integrals: If $0 \le x \le 1$, this area can be expressed as the sum of the area of a triangle and the area of a region under the unit circle: $\dfrac{x\sqrt{1 - x^2}}{2} + \int_{x}^{1} (\sqrt{1 - t^2}) dt$

2015年4月24日 · If theta corresponds to the point (x,y) on the unit circle, then: csc theta = 1/y sec theta = 1/x cot theta = x/y

By most definitions of dimension, however, the unit circle in $\mathbf{R^2}$ has dimension one. I understood the affine hull of unit circle has dimension two because the all of values of affine hull is in $\mathbf{R^2}$. But still I don't understand why the unit circle in $\mathbf{R^2}$ has one dimension. Thanks.

2019年12月27日 · An injective continuous self-map of the unit circle is a homeomorphism. 0.

Functions that maps unit circle into unit circle. Related. 2. Analytic continuation on the unit disc. 8.

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2018年5月24日 · Consider an arc AM, with extremity M that rotates counterclockwise on the trig unit circle, with radius R = 1 unity. The projection of arc AM on the 4 axis defines the 4 main trig functions: The projection of arc AM on the axis OA x defines f(x) = cos x The projection of AM on the axis OBy defines f(x) = sin x

I realize that I can consult integration tables for integrals like this, but I wanted to test my knowledge of integration techniques to solve this integral for the area of the unit circle. $$2\int_{-1}^1\sqrt{1-x^2}dx$$

As for linear transformations, if they have two eigenvalues, you end up "distorting" the unit circle along the corresponding eigenspaces. Take them orthogonal and you get ellipses. $\endgroup$ – Arturo Magidin

Students look at a circle as a $2$-D shape geometrically, and then don't get that topologically it can be described with a single parameter. $\endgroup$ – rschwieb Commented Jul 3, 2014 at 15:34