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Precalc/Trig

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Find the smallest positive number t such that e^sin(t)=1/2

  • Precalc/Trig -

    take ln of both sides

    ln(e^sin(t) = ln(1/2)
    sin(t) (lne) = ln 1 - ln 2 , but lne = 1 and ln 1 = 0

    sin(t) = -ln2 = -.69314718
    t must be in quads III or IV
    angle in standard position is 43.9° or .7658 radians

    in degrees,
    t = 180+43.9 = 223.9° or 360-43.9 = 316.1°

    in radians
    t = π+.7658 = 3.907 radians or 2π-.7658 = 5.517 radians

    check one of them
    t = 316.1°
    e^sin 316.1° = .4998 , not bad

    so the smallest value of t = 3.907

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