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  1. #1
    Member u2_wa's Avatar
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    Question roots of the equation

    Find the roots of the equations:
    1. sin(x)=x
    2. cos(x)=x
    3. tan(x)=x
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  2. #2
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    draw the graphs in each case to find solutions.

    suppose,y=sinx & y=x gives the solution of first equation.
    though sin(x) becomes x when x is a very small proper fraction.
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  3. #3
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    Quote Originally Posted by u2_wa View Post
    Find the roots of the equations:
    1. sin(x)=x
    2. cos(x)=x
    3. tan(x)=x
    These are transcendental equations - which means that exact solutions cannot be found except for special cases eg. For 1. and 3. x = 0 is one of the solutions. To get the other solutions you will need to use technology to get decimal approximations of them.

    Quote Originally Posted by sbcd90 View Post
    draw the graphs in each case to find solutions.



    suppose,y=sinx & y=x gives the solution of first equation.

    though sin(x) becomes x when x is a very small proper fraction.
    This is only good if either:

    1. You don't want very good accuracy in your solutions, or

    2. You use technology to draw them and then get that same technology to find the x-coordinates of the intersection points.
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  4. #4
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    e^(i*pi)'s Avatar
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    Quote Originally Posted by mr fantastic View Post
    These are transcendental equations - which means that exact solutions cannot be found except for special cases eg. For 1. and 3. x = 0 is one of the solutions. To get the other solutions you will need to use technology to get decimal approximations of them.
    You could also leave them in terms of \pi though?

    For example \sin{x} = 0 would have solutions of 0 + {k}\pi  where k is an integer


    Quote Originally Posted by sbcd90 View Post
    though sin(x) becomes x when x is a very small proper fraction.
    Note this only words if x is in radians, the small angle approximation (what sbcd90 describes) is a truncation of the Taylor series for sin, cos and tan respectively. It works because as x is much less than one it's powers will decrease rapidly.

    \sin{x} = x
    \cos{x} = 1-\frac{x^2}{2}
    \tan{x} = x
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  5. #5
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    Quote Originally Posted by e^(i*pi) View Post
    You could also leave them in terms of \pi though?

    For example \sin{x} = 0 would have solutions of 0 + {k}\pi where k is an integer

    [snip]
    x = k \pi is not a solution to \sin x = x. The only solution of this form is x = 0 .
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