Double Angle Formula
Since quaternions use expressions like sin(t/2) and cos(t/2) it would be useful to have expressions for these in terms of sin(t) and cos(t)
As a starting point take the following trig functions:
sin(2A) = 2 sin(A) cos(A)
cos(2A) = 2 cos²(A) - 1 = 1 - 2 sin²(A)
where:
- cos²(A) is shorthand notation for (cos(A))², that is, the square of cos
- sin²(A) is shorthand notation for (sin(A))², that is, the square of sine
Half Angle Formula
- sin(t/2) =√(0.5 (1- cos(t)))
- cos(t/2) =√(0.5 (1+ cos(t)))
- tan(t/2) = sin(t)/(1+cos(t))
Graphical Representation
We can show these relationships graphically
where the angle is shown at the centre of a
unit circle and the half angle is the angle at
a point on the circumference.
tan(θ)=opposite/adjacent = sin(θ)/(cos(θ)+1)
Derivation
In the above double angle formula we substitute t=2A to give:
1 - 2 sin²(t/2) = cos(t)
sin²(t/2) =0.5 (1- cos(t))
sin(t/2) =√(0.5 (1- cos(t)))
Similarly for cosine:
2 cos²(t/2) - 1 = cos(t)
2 cos²(t/2) = 1 + cos(t)
cos(t/2) =√(0.5 (1+ cos(t)))
