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Math Help - How many solutions in 4sin2x=3?

  1. #1
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    How many solutions in 4sin2x=3?

    Hello guys,
    I am just wandering how many solution there are to the equation in the title?
    The boundary is 0_<X_<360(degrees) (_< meaning less than or equal to sign)

    I worked out:
    sin2x=0.75
    2x=sin-1(0.75)
    2x=48.9o
    x=48.9x0.5
    x= 24.3

    I know this is one solution, but is there any more solutions to this?

    Thanks
    Maddie
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  2. #2
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    Re: How many solutions in 4sin2x=3?

    I believe there are four solutions in this range. An easy way to see this is to graph them. See the attached pdf.

    Do you know how to find the other solutions?
    Attached Files Attached Files
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  3. #3
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    Re: How many solutions in 4sin2x=3?

    Quote Originally Posted by MaddieMoodle View Post
    Hello guys,
    I am just wandering how many solution there are to the equation in the title?
    The boundary is 0_<X_<360(degrees) (_< meaning less than or equal to sign)

    I worked out:
    sin2x=0.75
    2x=sin-1(0.75)
    2x=48.9o
    x=48.9x0.5
    x= 24.3

    I know this is one solution, but is there any more solutions to this?

    Thanks
    Maddie
    The period of this function is \displaystyle \begin{align*} 180^{\circ} \end{align*}, and there are two solutions for each period.
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  4. #4
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    Re: How many solutions in 4sin2x=3?

    There are 4 solutions
    24.3 , 204.3 , 65.7 and 245.7
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  5. #5
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    Re: How many solutions in 4sin2x=3?

     0 \leq x \leq 360^{o} \Rightarrow 0 \leq 2x \leq 720^{o}

     sin(2x) = \frac{3}{4}

     2x = Arcsin(\frac{3}{4})

     2x = 360^{o} + Arcsin(\frac{3}{4})

     2x = 180^{o} - Arcsin(\frac{3}{4})

     2x = 540^{o} - Arcsin(\frac{3}{4})

     \Rightarrow

     x = \frac{Arcsin(\frac{3}{4})}{2} \approx 24.3^o

     x = \frac{180^{o} - Arcsin(\frac{3}{4})}{2} \approx 65.7^o

     x = \frac{360^{o} + Arcsin(\frac{3}{4})}{2} \approx 204.3^o

     x = \frac{540^{o} - Arcsin(\frac{3}{4})}{2} \approx 245.7^o
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