Maths - Trigonometry - Derived Trig Functions

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:

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))

half angle formula

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)))


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