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Since the circumference of the unit circle happens to be $ (2\pi)$, and since (in analytical geometry or trigonometry) this translates to $ (360^\circ)$, students new to calculus are taught about radians, which is a very confusing and ambiguous term. The generalization to higher dimensions is the unit sphere. By unit circle, i mean a certain conceptual framework for many important trig facts and properties, not a big circle drawn on a sheet of paper that has angles labeled with degree measures 30, 45, 60, 90, 120, 150, etc

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(and/or with the corresponding radian measures), along with the exact values for the sine and cosine of these angles. The unit circle is often denoted s1 Time is point rotation in a circle

There are 2 other circles and 2 other point rotations around those circles that are all mutually perpendicular to each other, therefore separate dimensions.

I do understand that the unit circle has a radius of 1 and sides of triangles made within it must pertain to the pythagorean theorem (hence these values with radicals, for accuracy), but that is all i understand How would one know to put exactly $\frac {\sqrt 3} {2}$ for the sine of $\frac {\pi} {3}$ radians This is unclear to me. Maybe a quite easy question

Why is $s^1$ the unit circle and $s^2$ is the unit sphere Also why is $s^1\\times s^1$ a torus It does not seem that they have anything. Show that unit circle is not homeomorphic to the real line ask question asked 7 years, 7 months ago modified 6 years, 2 months ago

In mathematics, the unit circle is a circle with a radius of one

Frequently, especially in trigonometry and geometry, the unit circle is the circle of radius one centered at the origin (0,0) in the cartesian coordinate system in the euclidean plane