Element-wise Formula for RoPE Rotation

Consider a system where positional information is encoded using periodic functions. A key component is a vector constructed for each position. Given a position index i = 2 and a set of frequency parameters θ = [π/2, π/4], what is the resulting 2-dimensional sine vector, defined as [sin(iθ₁), sin(iθ₂)]?

Consider a method for encoding positional information where a d-dimensional input vector is modified using periodic functions. In this method, a 'sine vector' is constructed for each position i. This sine vector has a dimensionality equal to d.

Determining Position Index from a Sine Vector