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Math Help - Exact values in radians of θ, goven cosec & tan

  1. #1
    Junior Member froodles01's Avatar
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    Exact values in radians of θ, goven cosec & tan

    Given cosecθ= -2, tanθ= 1/3√3 and -π < θ < π.
    I must find the exact value of θ in radians, justifying my answer.

    It shouldn't be too difficult;
    An answer is
    cosec theta = -2 implies sin theta = -(1/2); value of tan theta cannot be (1)/(3)√(3) but if it is taken as 1/√(3); then required theta = -(5*pi/6).

    Or is this incorrect?
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  2. #2
    Super Member bigwave's Avatar
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    Re: Exact values in radians of θ, goven cosec & tan

    this appears to be in Q3 not sure what you mean by -π < θ < π.

    both of these are \frac{-5\pi}{6} in Q3
    Last edited by bigwave; December 6th 2011 at 03:48 PM. Reason: added info, spelling
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  3. #3
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    Re: Exact values in radians of θ, goven cosec & tan

    Hello, froodles01!

    Your answer is correct!


    \csc\theta = \text{-}2,\;\tan\theta= \tfrac{\sqrt{3}}{3}\,\text{and }\,\text{-}\pi \,<\,\theta\,<\,\pi

    \text{Find the exact value of }\theta\text{ in radians.}

    \csc\theta \,=\,\text{-}2 \quad\Rightarrow\quad \sin\theta \,=\,\text{-}\tfrac{1}{2} \quad\Rightarrow\quad \theta \:=\:\text{-}\tfrac{5\pi}{6},\;\text{-}\tfrac{11\pi}{6}

    Since \tan\theta is positive, \theta is in Quadrant 1 or 3.

    Therefore: . \theta \:=\:-\frac{5\pi}{6}

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