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A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple ratios, such as 45:45:90. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3:4:5. Knowing the ratios of the angles or sides of these special right triangles allows one to quickly calculate various lengths in geometric problems without resorting to more advanced methods. Contents 1 Angle-based 1.1 45-45-90 triangle 1.2 30-60-90 triangle 2 Side-based 2.1 Common Pythagorean triples 2.2 Fibonacci triangles 2.3 Almost-isosceles Pythagorean triples 3 Right triangle whose angles are in a geometric progression 4 Right triangle whose sides are in a geometric progression 5 Calculating common trig functions 6 See also 7 External links 8 References // Angle-based "Angle-based" special right triangles are specified by the integer ratios of the angles of which the triangle is composed. The integer ratios of the angles of these triangles are such that the larger (right) angle equals the sum of the smaller angles: . The side lengths are generally deduced from the basis of the unit circle or other geometric methods. This form is most interesting in that it may be used to rapidly reproduce the values of trigonometric functions for the angles 30°, 45°, and 60°. 45-45-90 30-60-90 The 45-45-90 triangle, the 30-60-90 triangle, and the equilateral/equiangular (60-60-60) triangle are the three Möbius triangles in the plane, meaning that they tessellate the plane via reflections in their sides – see triangle group. 45-45-90 triangle The side lengths of a 45-45-90 triangle Constructing the diagonal of a square results in a triangle whose three angles are in the ratio With the three angles adding up to 180° (π), the angles respectively measure 45° (π/4), 45° (π/4), and 90° (π/2). The sides are in the ratio Here is a simple proof. Say you have such a triangle with legs a and b and hypotenuse c. Suppose that a = 1. Since two angles measure 45°, this is an isosceles triangle and we have b = 1. The fact that follows immediately from the Pythagorean theorem. Triangles with these angles are the only possible right triangles that are also isosceles triangles in Euclidean geometry. However, in spherical geometry and hyperbolic geometry, there are infinitely many different shapes of right isosceles triangles. 30-60-90 triangle The side lengths of a 30-60-90 triangle This is a triangle whose three angles are in the ratio and respectively measure 30°, 60°, and 90°. Since this triangle is half of an equilateral triangle, some refer to this as the hemieq triangle. The sides are in the ratio . The proof of this fact is clear using trigonometry. Although the geometric proof is less apparent, it is equally trivial: Draw an equilateral triangle ABC with side length 2 and with point D as the midpoint of segment BC. Draw an altitude line from A to D. Then ABD is a 30-60-90 (hemieq) triangle with hypotenuse of length 2, and base BD of length 1. The fact that the remaining leg AD has length follows immediately from the Pythagorean theorem. The hemieq triangle is the only right triangle whose angles are in an arithmetic progression. The proof of this fact is simple and follows on from the fact that if α, α+δ, α+2δ are the angles in the progression then the sum of the angles 3α+3δ = 180°. So one angle must be 60° the other 90° leaving the remaining angle to be 30°. Side-based All of the special side based right triangles possess angles that are never rational numbers of degrees[1], but whose sides are always of integer length and form a Pythagorean triple. They are most useful in that they may be easily remembered and any multiple of the sides produces the same relationship. Using Euclid's formula for generating Pythagorean triples, then the sides must be in the ratio where m and n are any positive integers such that m>n. Common Pythagorean triples There are several Pythagorean triples which are very well known, including those with sides in the ratios The 3:4:5 triangles are the only right triangles with edges in arithmetic progression. Triangles based on Pythagorean triples are Heronian, meaning they have integer area as well as integer sides. The following are all the Pythagorean triple ratios expressed in lowest form (beyond the five smallest ones, listed above) with both non-hypotenuse sides less than 256: Fibonacci triangles Starting with 5, every other Fibonacci number {0,1,1,2,3,5,8,13,21,34,55,89,...} is the length of the hypotenuse of a right triangle with integral sides, or in other words, the largest number in a Pythagorean triple. The length of the longer leg of this triangle is equal to the sum of the three sides of the preceding triangle in this series of triangles, and the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle. The first triangle in this series has sides of length 5, 4, and 3. Skipping 8, the next triangle has sides of length 13, 12 (5 + 4 + 3), and 5 (8 − 3). Skipping 21, the next triangle has sides of length 34, 30 (13 + 12 + 5), and 16 (21 − 5). This series continues indefinitely and approaches a limiting triangle with edge ratios: This right triangle is sometimes referred to as a dom, a name suggested by Andrew Clarke to stress that this is the triangle obtained from dissecting a domino along a diagonal. The dom forms the basis of the aperiodic pinwheel tiling proposed by John Conway and Charles Radin. Almost-isosceles Pythagorean triples Isosceles right-angled triangles cannot have sides with integer values. However, infinitely many almost-isosceles right triangles do exist. These are right-angled triangles with integral sides for which the lengths of the non-hypotenuse edges differ by one.[2] Such almost-isosceles right-angled triangles can be obtained recursively using Pell's equation: a0 = 1, b0 = 2 an = 2bn-1 + an-1 bn = 2an + bn-1 an is length of hypotenuse, n=1, 2, 3, .... The smallest Pythagorean triples resulting are: Right triangle whose angles are in a geometric progression The hemieq triangle above is the only right triangle whose angles are in an arithmetic progression. However there is another unique right triangle whose angles are in a geometric progression. The three angles are π/2φ2, π/2φ, π/2 where the common ratio is φ, the golden ratio. Consequently the angles are in the ratio The sides are in the ratio [3][citation needed] Because the sides are subject to Pythagoras' theorem this leads to the identity[citation needed] Interestingly, this can now be expanded into a phi identity that uses φ and the five fundamental mathematical constants π, e, i, 1, 0 of Euler's identity (though not as elegantly as the latter) as follows:[citation needed] Right triangle whose sides are in a geometric progression A Kepler triangle is a right triangle formed by three squares with areas in geometric progression according to the golden ratio. Main article: Kepler triangle The Kepler triangle is a right triangle whose sides are in a geometric progression. If the sides are formed from the geometric progession a, ar, ar2 then its common ratio r is given by r = √φ where φ is the golden ratio. Its sides are therefore in the ratio Calculating common trig functions Special triangles are used to aid in calculating common trig functions, as below: Degrees Radians sin cos tan 0 0 0 1 0 30 π / 6 1 / 2 √3 / 2 1 / √3 45 π / 4 1 / √2 1 / √2 1 60 π / 3 √3 / 2 1/2 √3 90 π / 2 1 0 - See also Triangle Integer triangle External links 3-4-5 triangle 30-60-90 triangle 45-45-90 triangle With interactive animations 45-45-90 triangle With explanations and examples References ^ Weisstein, Eric W. "Rational Triangle". MathWorld. http://mathworld.wolfram.com/RationalTriangle.html.  ^ C.C. Chen and T.A. Peng: Almost-isosceles right-angled triangles ^ Sloane, Neil J.A.. "Sequence A180014". On-line Encyclopedia of Integer Sequences. http://oeis.org/search?q=A180014.