Right triangles reflects the capability of depicting regular features, which leads to the easier state of calculations on triangles with the association of simpler formula is known as special right triangle. For instance, right triangles generally have simple relationships, 45°–45°–90°. The right triangles are also known as ‘angled-based right triangles’. In a ‘side based right triangle’, the sides reflects the length of whole number ratios, like 3:4:5 and also reflects special numbers like golden ratio. The understanding regarding the ‘ratios of the sides’ or the relationship amongst angles allows the simplification in calculation of special right triangles. The features of special right triangles enable quick calculation of lengths in the geometric interpretations. The angle-based special right triangles are depicted through relationship between angles that has formed the triangle. Special right triangle reflects angles that are larger right angles with the presence of 90 degree or in association with π/2 radians. In special right triangles, π/2 or 90 degrees equals the sum of rest two angles. Geometric methods or unit circle leads to the formation of side lengths of special right triangles. The approach of the special right triangle can be utilized for deducing the trigonometric functional values of triangles for angles like, 45 degree, 60 degree and 30 degree. Calculation of the trigonometric functions is availed by the features of special triangles.

Table: Depiction of trigonometric functions through special right triangle functions

Angled-based special triangles can be represented by relationship between the angles from which the special triangle is formed. In case of an angled-base special triangle, the largest angle that is π/2 radians or 90 degree reflects the sum of other two angles. The lengths of the special triangles are constructed from the basis of unit circle or from different geometric methods. The functionality and features of the special triangles are utilized for reproducing values regarding trigonometric functions. In case of the 45°–45°–90° triangle, constructing diagonal lines in square outcomes regarding triangles, where the angles reflect the ratio of 1:1:2 and summing up to 180 degree. The angles of this approach can be depicted as 45 degree (π/4), 90 degree (π/2) and 45 degree (π/4). Sides related to this approach of triangle reflect 1: 1: √2. In 45°–45°–90° triangle, smallest ration can be found, √2/2 as well as greatest ration regarding altitude of hypotenuse to sum of sides, √2/4. In case of the 30°–60°–90° triangle, three angles of the triangle represents the ration of 1:2:3. The angles of 30°–60°–90° triangle amounts, 30 degree (π/6), 90 degree (π/2) and 60 degree (π/3). The sides of this triangles depicts the ration of 1: √3: 2. The angles in 30°–60°–90° triangle portrays arithmetic progression.

In the side-based special right triangles, the sides of the triangles depicts length of integers and the composition of the length represents Pythagorean triplets and it reflects angles that cannot be rational number in respect to the degrees. The sides in the side-based special right triangle represents the formula deduced by Euclid regarding the Pythagorean triples can be represented as: m2-n2: 2mn: m2 + n2. In isosceles right triangle reflects the features of a right triangle and isosceles triangles. The right triangles have equal angles (two), equal sides (two) and a right angle, where the value of the right angle cannot be equal to the other two equal angles as it can exceed the sum of angles from 180 degrees. Special triangles reflects simplistic ratio between the sides of the each sides

In Pythagorean theorem, the calculation regarding the calculation of third side of the special right triangle. The two examples of the special right triangles are 45°–45°–90° and 30°–60°–90°, which are also used in the application of Pythagorean theorem. This approach contributes towards the identification of missing sides of special right triangles in the case of the available information regarding a single side length.

Figure: 30°–60°–90° special right triangle

The ratio of the sides of 30°–60°–90° special right triangle is x, 2x, x√3. The Pythagorean theorem can be confirmed through,

X² + (x√3)² = (2x)²

X² + 3x² = 4x²

4x² = 4x²

Figure: 45°–45°–90° special right triangle

The ratio of the sides of 45°–45°–90° special right triangle can be represented as x, x, x√2. The Pythagorean theorem can be confirmed through,

x² + x² = (x√2)²

2x² = 2x²

The examples of special right triangle are:

In case of the sides of a right triangle with sizes, x inches, 6 inches and 6 inches. The pattern in 45 degree-45 degree-90 degree special right triangles in association with the legs 6 inches and 6 inches reflects hypotenuse, which is 6 √2 inches. Therefore, x equals 6 √2.

In case of a special right triangle, 30°–60°–90° reflects hypotenuse of the length 10. The question is to find the size of rest two sides. Hypotenuse is the term used for the side opposite to 90-degree angle.

From the Table, the following equation can be formed:

10 = 2x

x = 5

x √3= 5 √3

Therefore, the two other sides deduced from the equation are, 5 √3

Identifying the missing length of the special right triangle in the form of a ratio.

Other two sides have length, 5 √2/2 and 5 √2/2.

x√2 = 5

x = 5 √2/2 * √2/√2 = 5√2/2

Therefore, the other side is 5√2/2