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Thread: Find all values of x in the interval

  1. #1
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    Find all values of x in the interval

    i can not figure out this math problem on my homework!! PLEASE HELP!


    Find all values of x in the interval [0,2π] that satisfy sin 2x= -sin x
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  2. #2
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    Hello, jpenny32!

    We need this double-angle identity: .$\displaystyle \sin2\theta \:=\:2\sin\theta\cos\theta$


    Find all values of $\displaystyle x$ in the interval $\displaystyle [0,\,2\pi]$ that satisfy: .$\displaystyle \sin 2x\:=\: -\sin x$
    Using the identity, we have: .$\displaystyle 2\sin x\cos x \:=\:-\sin x \quad\Rightarrow\quad 2\sin x\cos x + \sin x \:=\:0$


    Factor: .$\displaystyle \sin x(2\cos x + 1) \:=\:0$


    Then: . $\displaystyle \begin{array}{ccccccccccc}\sin x \:=\: 0 & \Rightarrow & \boxed{x \:=\: 0,\,\pi,\,2\pi} \\ \\[-3mm]
    2\cos x+1 \:=\: 0 & \Rightarrow & \cos x \:=\: -\frac{1}{2} & \Rightarrow & \boxed{x \:=\:\frac{2\pi}{3},\:\frac{4\pi}{3}} \end{array}$

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