Rotation of a Point using Bodhayana Triples

S. D. Mohgaonkar

Department of MathematicsShri Ramdeobaba College of

Engineering And Management, Nagpur

Three numbers a, b, c form a triple if a2 + b2 = c2 .

• If the angle in the triple is A then A)a, b, c denotesa right angled triangle with sides a, b, c :

• If a, b, c is a triple and p is a number then pa, pb, pc is also a triplet

A

a

bc

Bodhayana Triples:

Triples including Standard AnglesTriple

Included Angle in Degrees

Cos A Sin A Tan A

1, 0, 1 0 1 0 0

√3, 1, 2 30 √3/2 1/2 1/√3

1, 1, √2 45 1/√2 1/√2 1

1, √3, 2 60 1/2 √3/2 √3

0, 1, 1 90 0 1 ∞

-1, 0, 1 180

0, -1, 1 270

Addition and Subtraction of TriplesConsider Two triples A)x, y, z and B) X, Y, Z

Triple which contains the angle A+B is given byA x y zB X Y ZA+B xX- yY yX+xY zZ

Triple which contains the angle A-B is given byA x y zB X Y ZA+B xX+yY yX- xY zZ

Example: To find triple containing angle 1) 75o

30 √3 1 245 1 1 √275 √3-1 √3+1 2 √2

• Triple √3,1, 2 includes angle A= 300.• Triple 1,1, √2 includes angle B= 450.• Triple √3-1, √3+1, 2 √ 2 includes

angle A + B = 300 + 450= 750. Q(√3-1, √3+1)

Matrix of Rotation• Polar coordinates of a point P(x, y) are• x = r cosθ , y = r sin θ

• If a point (x, y) is rotated through an angle α in anticlockwise direction then it assumes new position P’ and its coordinates in new position are (x’, y’).• x’ = r cos (θ + α) = x cos α – y sin α• y’ = r sin (θ + α) = x sin α + y cos α

Rotation of a Point : Matrix of Rotation• Rotate a point (3, 4) through α = 300 in anticlockwise

direction

• Using above matrix of rotation, coordinates of new point are

Rotation of a Point : Bodhayana Triples• Rotate a point (3, 4) through α = 300 in anticlockwise

direction

• The point (3, 4) gives a right angled triangle 4, 3, 5• Triple corresponding to angle 300 is √3, 1, 2 which is same as 5

√3/2, 5/2, 5.• Rotation in anticlockwise direction => Addition of angles i.e.

addition of triples A 3 4 5 30 5 √3/2 5/2 5A+30 15√3/2 -20/2 15/2 + 20√3/2 25

(15√3 -20)/2 (15 + 20√3)/2 25 (3√3 - 4)/2 (3 + 4√3)/2 5

• Point in new position has coordinates ((3√3 - 4)/2, (3 + 4√3)/2)