Issue 125 - Sines and Cosines using Tirthaji’s Formula

Vedic Mathematics Newsletter No. 125
A warm welcome to our new subscribers.
This issue’s article, by K. Williams, is titled “Sines and Cosines using Tirthaji’s Formula”.

“The formula is easy enough to apply and gives surprisingly accurate results.”

This event will be on Saturday 14th March 2020.
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It is an international competition that tests ability and speed in using Vedic mathematics techniques and their application in problem solving. There are four levels for the Olympiad: Primary, Junior, Intermediate and Senior.

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EffiMath is an improvised adoption of Vedic Math. With deep roots of Vedic math methods and concepts, the learning is structured based on the grade level. Our focus is mainly on understanding concepts of Vedic Math (Learning books) and practicing the new procedures (Activity books). We also have modern certification and examination methods for the students to get certified. Our mission and vision is to empower children with the power of math in their life.


Sines and Cosines using Tirthaji’s Formula

Sri Bharati Krishna Tirthaji points out in his book [1] that there is Vedic material on ‘arcs and chords of circles, angles and sines of angles’. This is explained by some text on page 133 of a diary he wrote in [2]. On that page he gives a formula for calculating sines of angles given in radians. Though the formula looks complex it is easily adapted for cosines and for working in degrees.

A full description and explanation will be given at the upcoming Online Conference on 14th March so here we just show two examples.

The formula for degrees is: cosA = (8100-A^2) / (8100+0.25A^2)

So to find the cosine of 20 degrees we put A = 20 and get:

Cos 20 = (8100-20^2) / (8100+0.25x20^2)
    = (8100 - 400) /  (8100 + 100)
    = 7700 / 8200
    = 77 / 82
    = 0.9390

The formula is not exact. The difference between our answer here and the correct one is 0.0007.

Try it with a different angle. There is a division at the end of the calculation but this is easy with the Vedic straight division or recurring decimals method.

The sine of an angle is the cosine of the complementary angle, which means we can find a sine by just taking the given angle from 90 degrees and then applying the formula. For example:

Sine 50 = cos (90-50) = cos 40
    = (8100-40^2) / (8100+0.25x40^2)
    = 65 / 85
    = 13 / 17
    = 0.7647

The formula is easy enough to apply and gives surprisingly accurate results.

[1] B. K. Tirthaji, Vedic Mathematics, 1965, Motilal Banarsidass

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Editor: Kenneth Williams
The Vedic Mathematics web site is at:
11th January 2020


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