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Math Help - Hey Guys Please Help me solve this problem.... :)

  1. #1
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    Hey Guys Please Help me solve this problem.... :)

    So the question goes like this

    Solve the following trigonometric equation finding all solutions for X in the inerval [0, 2pi]

    2cos(2x)csc^2(x)=2cos(2x)



    Thanks in advance.
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  2. #2
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    Re: Hey Guys Please Help me solve this problem.... :)

    Hello, TrimHonduras!

    \text{Solve: }\:2\cos(2x)\csc^2(x) \:=\:2\cos(2x)\;\text{ for }x \in [0,\,2\pi]

    We have: . 2\cos(2x)\csc^2(x) - 2\cos(2x) \:=\:0

    Factor: . 2\cos(2x)\big[\csc^2(x)-1\big] \:=\:0


    We have two equations to solve:

    \cos(2x) \:=\:0 \quad\Rightarrow\quad 2x \:=\:\tfrac{\pi}{2},\:\tfrac{3\pi}{2},\:\tfrac{5\p  i}{2},\;\tfrac{7\pi}{2} \quad\Rightarrow\quad x \:=\:\tfrac{\pi}{4},\:\tfrac{3\pi}{4},\:\tfrac{5\p  i}{4},\:\tfrac{7\pi}{4}

    \csc^2(x)-1 \:=\:0 \quad\Rightarrow\quad csc^2(x) \:=\:1 \quad\Rightarrow\quad csc^2(x) \:=\:\pm1 \quad\Rightarrow\quad x \:=\:\tfrac{\pi}{2},\:\tfrac{3\pi}{2}
    Thanks from HallsofIvy
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  3. #3
    MHF Contributor

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    Re: Hey Guys Please Help me solve this problem.... :)

    Another way of looking at it: obviously if cos(x)= 0 then both sides are 0 so cos(x)= 0 gives solutions. If cos(x) is NOT 0, we can divide by 2cos(x) to get csc^2(x)= 1 so that csc(x)= 1 and csc(x)= -1 gives solutions.
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