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Math Help - Find sin2x for each given the following condition:

  1. #1
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    Find sin2x for each given the following condition:

    Find sin2x for each given the following condition:

    a) sin x = 4/5 for 90o < x < 180o

    b)tan x = -2 for 3pi/2 < x < 2pi
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  2. #2
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    Re: Find sin2x for each given the following condition:

    Find sin2x for each given the following condition:-codecogseqn-4.gif
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    Re: Find sin2x for each given the following condition:

    Hello, alejandro!

    \text{Find }\sin2x\text{ for each given the following conditions:}

    a)\;\sin x \,=\,\tfrac{4}{5}\:\text{ for }90^o < x < 180^o

    We are given: . \sin x \:=\:\tfrac{4}{5} \:=\:\tfrac{opp}{hyp}

    Angle x is in a right triangle with: opp = 3,\;hyp = 5.
    Pythagorus says: adj = \pm4 \quad\Rightarrow\quad \cos x \,=\,\pm\tfrac{4}{5}
    Since x in Quadrant 2, \cos x \,=\,-\tfrac{4}{5}

    Identity: . \sin2x \,=\,2\sin x\cos x

    Therefore: . \sin2x \;=\;2\left(\frac{3}{5}\right)\left(-\frac{4}{5}\right) \;=\; -\frac{24}{25}



    b)\;\tan x \,=\, -2\:\text{ for }\tfrac{3\pi}{2} < x < 2\pi

    Since x is in Quadrant 4, we have: . \tan x \,=\,\tfrac{\text{-}2}{1} \,=\,\tfrac{opp}{adj}

    Angle x is in a right triangle with: opp = \text{-}2,\;adj = 1
    Pythagorus says: . hyp \,=\,\sqrt{5} \quad\Rightarrow\quad \sin x \,=\,\text{-}\tfrac{2}{\sqrt{5}},\;\cos x \,=\,\tfrac{1}{\sqrt{5}}

    Therefore: . \sin 2x \:=\:2\left(-\frac{2}{\sqrt{5}}\right)\left(\frac{1}{\sqrt{5}} \right) \;=\; -\frac{4}{5}
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