Odd Angles Formula Trivia
01-22-2020, 12:45 AM
Post: #8
 Albert Chan Senior Member Posts: 901 Joined: Jul 2018
RE: Odd Angles Formula Trivia
(12-15-2019 01:49 AM)Albert Chan Wrote:  25 sin(x)6 = -cos(6x) + 6 cos(4x) - 15 cos(2x) + 20/2

Since even powers of sines expanded to sum of cosines + constant, we have:
Quote:$$\large \int _{-\pi \over 2} ^{\pi \over 2} (\sin x)^{2n} \;dx = \binom{2n}{n}\pi / 2^{2n}$$

Example, for perimeter of ellipse, where h = (a-b)/(a+b)

$$p = (a+b) \int _{-\pi \over 2} ^{\pi \over 2} \sqrt{(1 + h^2)+ 2h \sin x} \;dx$$

Let $$y={h \over 1+h^2}\;, z = 2y \sin x$$, and expand the integrand:

$$\sqrt{1 + h^2 + 2h \sin x} = \sqrt{(1+h^2)(1+z)} = \sqrt{1+h^2} \left[1 + \binom{½}{1}z + \binom{½}{2}z^2 + \binom{½}{3}z^3 + \binom{½}{4}z^4 + \binom{½}{5}z^5 + \binom{½}{6}z^6 +\;...\right ]$$

We can remove odd powers of z, since the area cancelled out.
Applying even powers of sine integral formula, we have:

$$p = \pi (a+b) \sqrt{1+h^2} \left[1 + \binom{½}{2}\binom{2}{1}y^2 + \binom{½}{4}\binom{4}{2}y^4 + \binom{½}{6}\binom{6}{3}y^6 +\; ... \right] = 2 \pi \sqrt{{a^2+b^2 \over 2}} \left[ 1 - {1\over 4}y^2 - {15\over 64}y^4 - {105 \over 256} y^6 -\; ... \right]$$

Unfortunately, convergence is much slower than series expansion on h.
However, we just proved $$p ≤ 2 \pi \sqrt{{a^2+b^2 \over 2}}$$

For a rough estimate, we can use Trapezoid's rule (2 trapezoids).

$$p ≈ \pi (a+b) \left({(1-h)\;+\; 2\sqrt{1+h^2}\;+\;(1+h) \over 4}\;\right) = \pi (a + b) \left({1 + \sqrt{1+h^2} \over 2}\;\right) = \pi \large\left( {a+b \over 2} + \sqrt{{a^2+b^2 \over 2}} \;\right)$$
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 Messages In This Thread Odd Angles Formula Trivia - Albert Chan - 12-20-2018, 05:38 PM RE: Odd Angles Formula Trivia - Albert Chan - 12-21-2018, 01:10 PM RE: Odd Angles Formula Trivia - Albert Chan - 08-14-2019, 01:54 PM RE: Odd Angles Formula Trivia - Albert Chan - 08-15-2019, 01:55 AM RE: Odd Angles Formula Trivia - Albert Chan - 08-15-2019, 03:42 PM RE: Odd Angles Formula Trivia - Albert Chan - 12-15-2019, 01:03 AM RE: Odd Angles Formula Trivia - Albert Chan - 12-15-2019, 01:49 AM RE: Odd Angles Formula Trivia - Albert Chan - 01-22-2020 12:45 AM

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