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The points labelled 1, Sec(θ), Csc(θ) represent the length of the line segment from the origin to that point. Sin(θ), Tan(θ), and 1 are the heights to the line starting from the x-axis, while Cos(θ), 1, and Cot(θ) are lengths along the x-axis starting from the origin.
It is possible to calculate the group velocity from the refractive-index curve n(ω) or more directly from the wavenumber k = ωn/c, where ω is the radian frequency ω = 2πf. Whereas one expression for the phase velocity is v p = ω/k, the group velocity can be expressed using the derivative: v g = dω/dk.
This is a list of Ig Nobel Prize winners from 1991 to the present day. [1]A parody of the Nobel Prizes, the Ig Nobel Prizes are awarded each year in mid-September, around the time the recipients of the genuine Nobel Prizes are announced, for ten achievements that "first make people laugh, and then make them think".