Element-wise Formula for RoPE Rotation

In a system that encodes positional information, a 'cosine vector' is defined for each position $i$ based on a set of frequency parameters $\theta = [\theta_1, \dots, \theta_{d/2}]$. The formula for this vector is: $[\cos i\theta_1, \dots, \cos i\theta_{d/2}]$.

Given a total dimension $d=4$, a position $i=2$, and frequency parameters $\theta_1 = 0.01$ and $\theta_2 = 0.0001$, what is the correct cosine vector for this position?

A developer is implementing a positional encoding scheme where a 'cosine vector' is needed for each position $i$. Given a set of frequency parameters $\theta = [\theta_1, \dots, \theta_{d/2}]$ and a total embedding dimension $d$, which of the following formulas correctly defines this vector?

Properties of Positional Cosine Vectors