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least positive value of csc(4 theta +3) sin( 5 theta - 6) =1

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  1. csc(4θ+3) sin(5θ-6) = 1
    That's surely an odd sort of expression. Still, since cscθ = 1/sinθ, we have

    sin(5θ-6) = sin(4θ+3)
    So, we want
    5θ-6 = 4θ+3
    θ = 9
    That seems kind of big, so what if we consider that sin(π-x) = sin(x). Then we have

    5θ-6 = π-(4θ+3)
    9θ = π+9
    θ = 1 + π/9
    Better, but we also know that
    sin(π+x) = -sin(x), so
    5θ-6 = -(π+(4θ+3))
    9θ = 3-π
    θ < 0 so we don't want that.

    sin(2π-x) = -sin(x), so
    5θ-6 = -(2π-(4θ+3))
    θ = 9-2π

    Hmmm. The problem is, that wolframalpha shows a solution at about θ=0.67 or roughly (1+π/9)/2. Maybe you can tweak the alternatives till you get that one.

    The graph is at

    http://www.wolframalpha.com/input/?i=solve+sin%285x-6%29+%3D+sin%284x%2B3%29+for+x+%3D+0..3

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    posted by Steve

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