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Thread: How do I solve this problem?

  1. #1
    Newbie
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    Nov 2008
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    How do I solve this problem?

    Solve each equation for x, where 0 ≤ Θ ≤ 2π

    cotx csc
    x = 2cotx

    the answer at the back says: π/4, π/2, 3π/4, 5π/4, 3π/2, or 7π/4

    How do I solve this?
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  2. #2
    Super Member

    Joined
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    Lexington, MA (USA)
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    Hello, lanvin!

    Exactly where is your difficulty?


    Solve for $\displaystyle x,\;0 \leq x < 2\pi\!:\quad\cot x \csc^2\!x \:= \:2\cot x$

    The answers are: .$\displaystyle \frac{\pi}{4},\;\frac{\pi}{2},\;\frac{3\pi}{4},\;\ frac{5\pi}{4},\;\frac{3\pi}{2},\;\frac{7\pi}{4}$

    We have: .$\displaystyle \cot x\csc^2\!x - 2\cot x \;=\;0$

    Factor: .$\displaystyle \cot x\bigg[\csc^2\!x - 2\bigg] \;=\;0 $


    And we have:

    . . $\displaystyle \cot x \:=\:0\quad\Rightarrow\quad\boxed{ x \:=\:\frac{\pi}{2},\;\frac{3\pi}{2}}$

    . . $\displaystyle \csc^2\!x \:=\:2\quad\Rightarrow\quad \csc x \:=\:\pm\sqrt{2} \quad\Rightarrow\quad\boxed{ x\:=\:\frac{\pi}{4},\;\frac{3\pi}{4},\;\frac{5\pi} {4},\;\frac{7\pi}{4}}$

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