In certain places, Vedic Mathematics talks about trigonometrical triplets, which became famous as Pythagorean triplets much later. So was the outcome of Pythagorean theorem known to Indians?
Without going into that discussion, let us see how this concept of triplets can be extremely effective in solving many types of trigonometric problems.
Let us define a triplet in a triangle as a (i.e. base), b (i.e. perpendicular) , c (hypotenuse). Here, ‘a’ and ‘b’ are the measures of the two sides followed by the hypotenuse (‘c’) at the end.
The relationship c2 = a2 + b2 holds true in this case.
Thus if any two values of the triplet are given, we can compute the 3rd value and build the complete triplet. Let us see some examples of how to build and use the triplet.
Case 1: suppose that a = 3 and b = 4. Let us see how to build triplet.
Given partial triplet is 3, 4, ______
Therefore the last value will be = √ (32 + 42) = 5. Hence the completed triplet is 3, 4, 5.
Case 2: If we’re given an incomplete triplet as 12, ___, 13. The missing value will be √ (132 – 122) = 5. Hence, the completed triplet would be 12, 5, 13.
Once the triplet is built, all the six trigonometric ratios can be read off easily without any further computation or use of any formulae.
Computing Trigonometric Ratios
If tan A = 4/3, find Cosec A.
Normally to find this we will use the formula -
Cosec2 A = 1 + Cot2 A
as, tan A = 4/3, cot A = ¾
On substituting this value in the given formula, we get
Cosec2 A = 1 + 9/16
and cosec A = 5/4, sin A = ⅘
Let us now see how to use triplets method of Vedic Maths to find cosec A and avoid various steps shown above.
The incomplete triplet in this example is
3, 4, ___
tan A = 4/3 = p / b
The complete triplet would be 3, 4, 5.
Hence, cosec A = h / p = 3/5
Isn’t this simply amazing? There’s more to come. We can also calculate double-angle and half-angle using similar method which we will discuss in a future post based on your interest. Let me know if it helps you!