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Math Help - Help with solving a trigometric equation.

  1. #1
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    Help with solving a trigometric equation.

    Hello,

    The equation is as follows:
    \cos 2\left( x_{+\frac{\pi }{8}} \right)=-\left( \frac{1}{\sqrt{2}} \right)

    With the interval \left[ 0,2\pi  \right].

    So what I did was let 2x+\frac{\pi }{4}=\frac{3\pi }{4} and 2x+\frac{\pi }{4}=\frac{5\pi }{4} since cos is negative in the 2nd and 3rd quadrants with a basic angle of \frac{\pi}{4}.

    I then continued to solve the two equations and ended up with x= \frac{\pi }{4} and \frac{\pi }{2}.

    However I'm missing out the two angles \frac{5\pi }{4} and \frac{3\pi }{2}. I'm supposed to be adding 2\pi to \frac{\pi }{4} and \frac{\pi }{2} right? But it doesn't give me the other two angles that I missed out.

    Also, regarding intervals, when I multiply the a number into the brackets, does it change? Like for the above when I multiply the 2 to get 2x, does the interval get multiplied by two also? That is \left[ 0,4\pi  \right].
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  2. #2
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    Quote Originally Posted by n0d3 View Post
    Hello,

    The equation is as follows:
    \cos 2\left( x_{+\frac{\pi }{8}} \right)=-\left( \frac{1}{\sqrt{2}} \right)

    With the interval \left[ 0,2\pi  \right].
    \cos\left(2x+\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}}

    0 \le x \le 2\pi

    0 \le 2x \le 4\pi

    \frac{\pi}{4} \le 2x + \frac{\pi}{4} \le \frac{17\pi}{4}

    2x+ \frac{\pi}{4} = \frac{3\pi}{4} \, , \, \frac{5\pi}{4} \, , \, \frac{11\pi}{4} \, , \, \frac{13\pi}{4}
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  3. #3
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    Thanks, kinda made it clearer.
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  4. #4
    Senior Member pacman's Avatar
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    Nice solution skeeter, the graph checks it.

    x1 = (pi/2)(2k - 1), k is an element of Z

    x2 = (pi/4)(4k + 1), k is an element of Z
    Attached Thumbnails Attached Thumbnails Help with solving a trigometric equation.-pi.gif   Help with solving a trigometric equation.-pi-12.gif  
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  5. #5
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    So I think my mistake would be that I should have treated the entire equation and find all the angles within the domain first before solving for x. What I did was solve for x then add/minus the 2pi to get the other angles, which was wrong.
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