Neumann s triangle is consistent with the geometrical restriction that \ alpha + \ beta + \ theta = 2 \ pi, and applying the law of sines and law of cosines to it produce relations that describe how the interfacial angles depend on the ratios of surface energies.
42.
But, given that, and that everything's in a plane, you can do it using nothing but the law of cosines, and the rule that all the angles in sequence around a point must add to 0 mod 2pi . talk ) 19 : 22, 28 April 2008 ( UTC)
43.
Isn't it redundant because it follows directly from the law of cosines, since the law of cosines gives the " unique " possible value for the third side of an angle given two sides and the angle between them ?-- talk ) 22 : 18, 28 November 2009 ( UTC)
44.
Isn't it redundant because it follows directly from the law of cosines, since the law of cosines gives the " unique " possible value for the third side of an angle given two sides and the angle between them ?-- talk ) 22 : 18, 28 November 2009 ( UTC)
45.
The fifth paper is " Elliptic and Hyperbolic Analysis " which considers the spherical law of cosines as the fundamental theorem of the sphere, and proceeds to analogues for the ellipsoid of revolution, general ellipsoid, and equilateral hyperboloids of one and two sheets, where he provides the hyperbolic law of cosines.
46.
The fifth paper is " Elliptic and Hyperbolic Analysis " which considers the spherical law of cosines as the fundamental theorem of the sphere, and proceeds to analogues for the ellipsoid of revolution, general ellipsoid, and equilateral hyperboloids of one and two sheets, where he provides the hyperbolic law of cosines.
47.
Finally, the distance between Earth and Jupiter can be calculated using standard trigonometry, in particular the law of cosines, knowing two sides ( distance between the Sun and Earth; distance between the Sun and Jupiter ) and one angle ( the angle between Jupiter and Earth as formed at the Sun ) of a triangle.
48.
These identities reduce to the cosine rule of plane trigonometry in the limit of sides much smaller than the radius of the sphere . ( On the unit sphere " a, b, c \ sin a \ approx a and \ cos a \ approx 1-a ^ 2 / 2 etc .; see Spherical law of cosines .)
49.
:The significance of the law of sines and the law of cosines is that you can use them to solve a triangle if you know two sides and an angle ( law of cosines ) or two angles and a side ( law of sines ) . & # x2013; b _ jonas 12 : 13, 28 March 2006 ( UTC)
50.
:The significance of the law of sines and the law of cosines is that you can use them to solve a triangle if you know two sides and an angle ( law of cosines ) or two angles and a side ( law of sines ) . & # x2013; b _ jonas 12 : 13, 28 March 2006 ( UTC)
