i am reading Lillian R. Lieber's book on the einstein theory of relativity and i am a bit confused on page 65. she wants to take the equations: x=x'cosθ - y'sinθ y=x'sinθ + y'cosθ and compare them to: x'=β(x-vt) t'=β(t-vx/c2) she takes c as one so: x'=β(x-vt) t'=β(t-vx) she solves for x and t and gets: x=β(x'+vt') t=β(t'+vx') then she replaces t with iτ and t' with iτ' and she gets: x=β(x'+vt') iτ = iβτ' + βvx' x=β(x'+vt') τ=βτ' + iβvx' next is the part i am confused about. she sets β = cosθ and -iβv = sinθ. this nicely turns the Lorentz equations into: x=x'cosθ - τ'sinθ τ=x'sinθ + τ'cosθ what i don't understand is how did she choose -iβv = sinθ? it works out all nicely in the end but how did she know that sinθ had to equal -iβv? was it arbitrary as a result of trial and error or did she use β = cosθ in order to figure out that -iβv = sinθ?