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Since the area of a circular sector with radius r and angle u (in radians) is r 2 u/2, it will be equal to u when r = √ 2. In the diagram, such a circle is tangent to the hyperbola xy = 1 at (1,1). The yellow sector depicts an area and angle magnitude.
Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin( α + β ) = sin α cos β + cos α sin ...
For the sine function, we can handle other values. If θ > π /2, then θ > 1. But sin θ ≤ 1 (because of the Pythagorean identity), so sin θ < θ. So we have < <. For negative values of θ we have, by the symmetry of the sine function
The quantity 206 265 ″ is approximately equal to the number of arcseconds in a circle (1 296 000 ″), divided by 2π, or, the number of arcseconds in 1 radian. The exact formula is = (″) and the above approximation follows when tan X is replaced by X.
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For arcoth, the argument of the logarithm is in (−∞, 0], if and only if z belongs to the real interval [−1, 1]. Therefore, these formulas define convenient principal values, for which the branch cuts are (−∞, −1] and [1, ∞) for the inverse hyperbolic tangent, and [−1, 1] for the inverse hyperbolic cotangent.
Equivalently, is conjugate to in if and only if and satisfy the Cauchy–Riemann equations in . As an immediate consequence of the latter equivalent definition, if is any harmonic function on , the function is conjugate to for then the Cauchy–Riemann equations are just = and the symmetry of the mixed second order derivatives, =.
Fig. 1: The Ming Antu Model Fig. 3: Ming Antu independently discovered Catalan numbers.. Ming Antu's infinite series expansion of trigonometric functions.Ming Antu, a court mathematician of the Qing dynasty did extensive work on the infinite series expansion of trigonometric functions in his masterpiece Geyuan Milü Jiefa (Quick Method of Dissecting the Circle and Determination of The Precise ...