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In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle.It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.
In this way, this trigonometric identity involving the tangent and the secant follows from the Pythagorean theorem. The angle opposite the leg of length 1 (this angle can be labeled φ = π/2 − θ) has cotangent equal to the length of the other leg, and cosecant equal to the length of the hypotenuse. In that way, this trigonometric identity ...
Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin( α + β ) = sin α cos β + cos α sin ...
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.
The top example shows a case where z is much less than the sum x + y of the other two sides, and the bottom example shows a case where the side z is only slightly less than x + y. In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the ...
However, any Pythagorean triangle with legs x, y and hypotenuse z can generate a Pythagorean triangle with an integer altitude, by scaling up the sides by the length of the hypotenuse z. If d is the altitude, then the generated Pythagorean triangle with integer altitude is given by [ 11 ]
By the Pythagorean theorem, the sum of the squares of the lengths of the catheti is equal to the square of the length of the hypotenuse. The term leg, in addition to referring to a cathetus of a right triangle, is also used to refer to either of the equal sides of an isosceles triangle or to either of the non-parallel sides of a trapezoid.
The Kepler triangle is a right triangle whose sides are in geometric progression. If the sides are formed from the geometric progression a, ar, ar 2 then its common ratio r is given by r = √ φ where φ is the golden ratio. Its sides are therefore in the ratio 1 : √ φ : φ. Thus, the shape of the Kepler triangle is uniquely determined (up ...