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How does e, or the exponential function, relate to rotation? First, assume the Unit Circle Parameter is Time in Seconds The essential idea is that in order for a Radius of Length 1 to move 1 Arc Length in 1 Second it is required to have a Velocity of 1, Acceleration of 1, Jolt of 1, etc
calculus - Trigonometric functions and the unit circle - Mathematics . . . 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
How do I get the slope on a circle? - Mathematics Stack Exchange The prior answers have all used calculus I'm going to post an answer using only trig The following diagram from Wikipedia's Trig Page is helpful However, that diagram also has a fault--the picture is very cluttered :) Thus, I've redrawn it for you, labeling the components important for this problem: Note that $\csc\theta$ returns the distance from the origin to the y-intercept of the
Möbius transformation mapping - Mathematics Stack Exchange 7 For your first example, because the boundary of the upper half-plane is a "circle" (in the Riemann sphere sense (sorry, Riemann sphere, not Bloch sphere)), and the boundary of the unit disk is a circle (plainly, but also in the Riemann sphere sense), we try to map the boundary of the one to the boundary of the other