1. Rotation of a Point using Bodhayana Triples
S. D. Mohgaonkar
Department of Mathematics
Shri Ramdeobaba College of Engineering
And Management, Nagpur
2. 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 denotes
a 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
b
c
Bodhayana Triples:
3. Triples including Standard Angles
Triple
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
4. Addition and Subtraction of Triples
Consider Two triples A)x, y, z and B) X, Y, Z
Triple which contains the angle A+B is given by
A x y z
B X Y Z
A+B xX- yY yX+xY zZ
Triple which contains the angle A-B is given by
A x y z
B X Y Z
A+B xX+yY yX- xY zZ
5. Example: To find triple containing angle 1) 75o
30 √3 1 2
45 1 1 √2
75 √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)
6. 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
7. 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
8. 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 5
A+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)
