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30MSC 2023 #29:Triples.

30MSC 2023 #29:Triples.

A triple is a set of three numbers which satisfies the Pythagorean Theorem: a^2 + b^2 = c^2, that is, the sum of the squares of the first two numbers is equal to the square of the third.

Whereas, in traditional trigonometry, angles are measured in degrees or radians while in Vedic Mathematics, they are described as triples or sides of a right triangle.  The first figure is the base (or adjacent side); the second is the height (or opposite side) and the third figure is the hypotenuse of the right triangle.

If an angle is expressed in terms of the measures of the sides of the right triangle, then there is no need to use tables to find the trigonometric ratios.

And if we can have a procedure to combine two triples to produce another triple, then we can do away will all those addition, subtraction and double angle formulas we are required to memorize in traditional trigonometry.

The solution to our featured example for today which was taken from the Senior IVMO 2021 shows how an addition two triples will produce another triple: Given two triples, A) 3 4 5 and B) 24 7 25, what is the triple for A + B?

The procedure to add these two triples is:

  1. Multiply the first elements and subtract the product of the second elements to get the first figure of the result: (3 x 24) – (4 x 7) = 44
  2. Cross-multiply the first and second elements and add their products to get the second element of the result: (3 x 7) + (4 x 24) = 117
  3. The product of the 3rd figures is the 3rd figure of the angle addition: 5 x 25 = 125

A                            3                            4                            5

B                            24                          7                            25                   

          A + B          [(3 x 24) – (4 x 7)]    [(3 x 7) + (4 x 24)]    (5 x 25)

             =                           44                         117                        125  

Many other applications of triples are presented in Chapter 9 of our forthcoming book, 30 Master Strategies in Computing, Vol III.

Follow us on our Facebook page, MATH-Inic Philippines at https://www.facebook.com/MATHInicPhils to see tomorrow’s 30MSC 2023 #30: By Mere Observation and all of our MATH Specials for Christmas.

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