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Math Help - Solving for an angle involving cos and sin

  1. #1
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    Solving for an angle involving cos and sin

    This problem is part of the analysis of a statics problem.

    There is a force P applied to a system which is 0.589kN. I need to determine if there is such an angle that would allow us to apply 86.6% of P and still be able to cut through a board that requires 3.5kN of force to cut.

    All simplified I end up with this:

    3.5 = 3.03cos x + 2.04sin x

    How do I solve for x? I think you take the derivative of the expression and then solve.

    2.04 / 3.03 = sinx / cosx

    Is this the correct approach?
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  2. #2
    MHF Contributor ebaines's Avatar
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    You seem to have left out part of the problem, so please clarify:

    Where did the values 3.03 and 2.04 come from?

    Why do you think that taking a derivative is necessary?
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  3. #3
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    The values came from a previous part in the problem. It's way too long to post on here. And I believe you take the derivative because sin/cos is oscillating and I guess I want to find the angle which creates a maximum point on a graph. This maximum will have a slope of 0, hence taking the derivative gives 0 on one side and the expression on the other.

    I tried it out and got an angle which gave me a number very close to the 3.5 so I think I am doing it correctly. Something where x = around 33 degrees works well.


    If you do not think this is how you are supposed to solve it please clarify.
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  4. #4
    MHF Contributor ebaines's Avatar
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    I'll take yor word that you're trying to solve the equatoin:

    3.5 = 3.03 cos(x) + 2.04 sin(x)

    Note that you're not trying to find the max value of 3.03 cos(x) + 2.04 sin(x), but rather find where it equals 3.5, right? So again, no need to take a derivative.

    I find approximate solutions for x is 0.303 and 0.883 radians, or about 17.4 and 50.6 degrees.
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  5. #5
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    Quote Originally Posted by ebaines View Post
    I find approximate solutions for x is 0.303 and 0.883 radians, or about 17.4 and 50.6 degrees.
    How did you solve the equation? I feel like it's something really simple that I am missing.
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  6. #6
    MHF Contributor ebaines's Avatar
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    I simply made a graph of the function f(x) = 3.5 -(3.03cos x + 2.04sin x) using Excel and saw where it crosses the x axis. A more "mathematically rigorous" approach that you might find in a text book is to use Newton's method to zero in on more precise answers. With this method you make a guess at the solution, see how much error there is, make another guess, see how the amount of error has changed, then use that data to make the next guess, always trying to getteh error to zero.. You can iterate to a pretty precise answer in just a few steps. But with both techniques you don't get a closed-form solution, but rather just very precise estimates.
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  7. #7
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    3.5 = 3.03cos x + 2.04sin x
    You can solve it directly.
    3.5 - 3.03cosx = 2.04sinx
    Square both sides.
    12.25 + 9.18cos^2(x) - 21.21cos(x) = 4.16sin^2(x) = 4.16[1-cos^2(x)]= 4.16 - 4.16*cos^2(x)
    13.34*cos^2(x) - 21.21*cos(x) + 8.09 = 0.
    Solve the quadratics and find the value of cos(x). From that find x.
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  8. #8
    MHF Contributor ebaines's Avatar
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    Quote Originally Posted by sa-ri-ga-ma View Post
    3.5 = 3.03cos x + 2.04sin x
    You can solve it directly.
    3.5 - 3.03cosx = 2.04sinx
    Square both sides.
    12.25 + 9.18cos^2(x) - 21.21cos(x) = 4.16sin^2(x) = 4.16[1-cos^2(x)]= 4.16 - 4.16*cos^2(x)
    13.34*cos^2(x) - 21.21*cos(x) + 8.09 = 0.
    Solve the quadratics and find the value of cos(x). From that find x.
    Excellent!
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