The Pythagorean theorem, and hence this length, can also be derived from the law of cosines by observing that the angle opposite the hypotenuse is 90 & deg; and noting that its cosine is 0:
32.
In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles a consequence of the law of cosines, although it may be proven without these theorems.
33.
A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, given the lengths of the other two sides and the angle between them.
34.
The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known, and in computing the angles of a triangle if all three sides are known.
35.
Equivalently, the law of cosines may be used to first compute the length of the spoke as projected on the wheel's plane ( as illustrated in the diagram ), followed by an application of the Pythagorean theorem.
36.
A generalization of this theorem is the law of cosines, which allows the computation of the length of the third side of any triangle, given the lengths of two sides and the size of the angles between them.
37.
Here two cases of non-Euclidean geometry are considered spherical geometry and hyperbolic plane geometry; in each case, as in the Euclidean case for non-right triangles, the result replacing the Pythagorean theorem follows from the appropriate law of cosines.
38.
I got to thinking about this topic by considering that there are only a few angles with rational degree that also have rational trigonometric functions; and the fact that the law of cosines is very similar the Pythagorean relationship.
39.
Rapidities \ mathbf w _ 1, \ mathbf w _ 2 with directions inclined at an angle \ theta have a resultant norm w \ equiv | \ mathbf w | ( ordinary Euclidean length ) given by the hyperbolic law of cosines,
40.
The law of tangents, although not as commonly known as the law of sines or the law of cosines, is equivalent to the law of sines, and can be used in any case where two sides and the included angle, or two angles and a side, are known.
