The plot of the equation cos(x+y)=cos(x)+cos(y) looks like a collection of infinitely many identical closed shapes.
Find the area of one of these closed shapes.
If in fact the shape is an ellipse, as it appears to be when plotting in Wolfram Alpha, it looks as if each "cup" is an ellipse with a minor axis either on or parallel to y=x.
Solving, it's easy enough to find the length of the minor axis:
The first solution in the first quadrant of the equality intersecting the line y = x is at
x = y = 2 arctan(sqrt(1 + 2/sqrt(3)))
the second is at
x = y = 2 * pi - 2 arctan(sqrt(1 + 2/sqrt(3)))
The latter minus the former is
2 * pi - 4 arctan(sqrt(1 + 2/sqrt(3))) ~= 2.39212378817231
Since this is the difference in the x values, but lies on the line y = x, the distance between these is that value times sqrt(2):
(2 * pi - 4* atan(sqrt(1 + 2/sqrt(3)))) * sqrt(2) ~= 3.3829739041086
For the major axis:
The center of this ellipse (the one in the first quadrant and nearest the origin) is thus, thanks to all those coefficients of 2 just at x = y = pi, and since the major axes have slope -1, lie on the line y = 2*pi - x.
The two endpoints of the major axis have x coordinates:
5 * pi /3 and pi / 3, the difference being 4 * pi/3
Again since it's sloped at a 45° angle we multiply by sqrt(2) getting
(4*pi/3) * sqrt(2) ~= 5.92384391754452 as the major axis
The area is pi * R * r:
approximately 3.3829739041086 * 5.92384391754452 * pi/4
with the division by 4 because we used major and minor diameters rather than radii.
And the approximate area is thus 15.7395436451308.
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Posted by Charlie
on 2020-08-07 12:09:02 |
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