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A Pythagorean triple has three positive integers a, b, and c, such that a 2 + b 2 = c 2. In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths. [1] Such a triple is commonly written (a, b, c). Some well-known examples are (3, 4, 5) and (5, 12, 13).
When d is chosen such that d = a/3, it generates a right triangle that is always similar to the Pythagorean triple with sides 3, 4, 5. Now consider a triangle whose sides are in a geometric progression and let the sides be a, ar, ar 2. Then the triangle inequality requires that
Two sides and an angle not included between them (SSA), if the side length adjacent to the angle is shorter than the other side length. A side and the two angles adjacent to it (ASA) A side, the angle opposite to it and an angle adjacent to it (AAS). For all cases in the plane, at least one of the side lengths must be specified.
Chapter 13 relates Pythagorean triangles to rational points on a unit circle, Chapter 14 discusses right triangles whose sides are unit fractions rather than integers, and Chapter 15 is about the Euler brick problem, a three-dimensional generalization of Pythagorean triangles, and related problems on integer-sided tetrahedra.
The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae , it is one of the basic relations between the sine and cosine functions.
The oldest and most elementary definitions are based on the geometry of right triangles and the ratio between their sides. The proofs given in this article use these definitions, and thus apply to non-negative angles not greater than a right angle. For greater and negative angles, see Trigonometric functions.
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