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Math Help - Need help solving for x!

  1. #1
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    Need help solving for x!

    sin(x) = sin(x + pi/3)

    I know this is simple but so am i... thanks in advance for any help
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  2. #2
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    e^(i*pi)'s Avatar
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    Quote Originally Posted by jsel21 View Post
    sin(x) = sin(x + pi/3)

    I know this is simple but so am i... thanks in advance for any help
    It seems painfully obvious that there is no value of x for which these two values would be equal. It is like asking when does 2 equal 2+1
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  3. #3
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    but there are solutions. When entering both of these in my graping calculator there are multiple points in which the two functions intersect. I just dont know how to solve for them. I'm looking for the the solutions between 0 and 2pi.
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  4. #4
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    Quote Originally Posted by e^(i*pi) View Post
    It seems painfully obvious that there is no value of x for which these two values would be equal. It is like asking when does 2 equal 2+1
    I know I posted this a while ago but I just wanted to point out that there are definitely solutions to this. x = pi/3, 4pi/3. I know how to solve in my head but not on paper.
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  5. #5
    Behold, the power of SARDINES!
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    Use the identity that

    \sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)

    This gives the equation

    \sin(x)=\frac{1}{2}\sin(x)+\frac{\sqrt{3}}{2}\cos(  x)

    Or

    \tan(x)=\sqrt{3}

    This will give you the solutions you want.
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  6. #6
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    If x is a solution then \sin(x) and \sin(x+\pi/3) must lie at the intersections of some horizontal line through the unit circle. Furthermore, the angle between these points (measured from the origin) must be \pi/3. But the other angles between the rightmost point and the positive x axis and the leftmost point and the negative x axis must be equal by symmetry, so call them \theta.

    This leads to the equation 2\theta+\pi/3=\pi, since together the three angles make a straight line. This gives you one solution and then the other is easily found.
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  7. #7
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    Quote Originally Posted by jsel21 View Post
    sin(x) = sin(x + pi/3)

    I know this is simple but so am i... thanks in advance for any help
    Visually, you could use the Unit Circle.

    sin(x) gives the y co-ordinate of a point on the Unit Circle,
    hence the y-axis from -1 to 1 acts as an axis of symmetry.

    0\le\ x\ \le \pi\Rightarrow\ sin(x)=sin\left(x+60^o\right)

    requires that x and x+60^0 are 30^o either side of the y-axis.

    x=90^o-30^o=60^o


    The exact same logic may be used underneath the x-axis for \pi\le\ x\ \le\ 2\pi

    to obtain a 2nd solution.

    Complete the analysis by adding multiples of 360 degrees to the two solutions.
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