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)


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)



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)))

metadata block
see also:


Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

cover Mathematics for 3D game Programming - Includes introduction to Vectors, Matrices, Transforms and Trigonometry. (But no euler angles or quaternions). Also includes ray tracing and some linear & rotational physics also collision detection (but not collision response).

Other Math Books

This site may have errors. Don't use for critical systems.

Copyright (c) 1998-2023 Martin John Baker - All rights reserved - privacy policy.