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- 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
- Tips for understanding the unit circle - Mathematics Stack Exchange
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 (and or with the corresponding radian measures), along with the exact values for the sine and cosine of these angles
- On Cotangents, Tangents, Secants, And Cosecants On Unit Circles.
Above is a diagram of a unit circle While I understand why the cosine and sine are in the positions they are in the unit circle, I am struggling to understand why the cotangent, tangent, cosecant,
- general topology - Why do we denote $S^1$ for the the unit circle and . . .
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
- calculus - Prove that the unit circle is path-connected? - Mathematics . . .
For proving that the unit circle is connected, you could also say that "the only subsets of the unit circle which are both open and closed are the full circle and the empty set"
- How does $e^ {i x}$ produce rotation around the imaginary unit circle?
Related: In this old answer, I describe Y S Chaikovsky's approach to the spiral using iterated involutes of the unit-radius arc The involutes (and spiral segments) are limiting forms of polygonal curves made from a family of similar isosceles triangles; the proof of the power series formula amounts to an exercise in combinatorics (plus an
- 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
- projection - Mapping a hemisphere onto the unit circle - Mathematics . . .
1 How can I approach attempting to map a hemisphere onto the unit circle such that the meridians become arcs of a circle through $ (0,1)$ and $ (0,-1)$ and are evenly spaced around the x-axis (which is mapped from the equator) and the lines of constant latitude also becomes arcs of a circle evenly spaced along the y-axis?
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