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Thread: Finding solutions of an equation

  1. #1
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    Finding solutions of an equation

    Find all of the solutions of the equation

    cos(56-3x)=-0.785

    where 0
    ≤ x ≤ 360


    I've been able to find one soloution using
    x = (56 - arccos(-0.785)) / 3. But I'm unable to find any others.

    Thanks in advance for any help
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  2. #2
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    Re: Finding solutions of an equation

    Since cos is oscillating function, $\cos(x)=a$ has infinite solutions.
    For any integer $n$, due to the fact that $\cos(2n\pi\pm\theta)=\cos(\theta)$, all the solutions of above equation can be written by, $x = 2n\pi \pm \arccos(a)$.
    Therefore, your problem of $\cos(\theta-kx)=a=\cos(kx-\theta)$ has solutions in the form of $x = \frac{2n\pi \pm \arccos(a) + \theta}{k}$ for all integer $n$.
    [Note that $\theta$ should be in Radians, not degrees]
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  3. #3
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    Re: Finding solutions of an equation

    Do you know what the graph of y= cos(x) looks like? cos(360- x)= cos(x). The principle solution to cos(x)= 0.785, in degrees, is 38.3 degrees. So 360- 38.3= 321.7 degrees is the other solution between 0 and 360 degrees. With 56- 3x= 321.7, x= (321.7- 56)/(-3)= -88.6 degrees.

    (zemozamster, the original problem is given in degrees so the answer should be in degrees.)
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  4. #4
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    Re: Finding solutions of an equation

    [QUOTE=(zemozamster, the original problem is given in degrees so the answer should be in degrees.)[/QUOTE]
    That is why the answer is given in a generic form. Radians to degree conversion is something that he/she require to have before coming to this type of problems.
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  5. #5
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    Re: Finding solutions of an equation

    Quote Originally Posted by zemozamster View Post
    Be patient, TEX will load, eventually...
    A bit like: posting "quick reply", please wait !!
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  6. #6
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    Re: Finding solutions of an equation

    cos(56-3x)=-0.785

    where 0 ≤ x ≤ 360


    cosine is an even function $\implies \cos(56-3x) = \cos(3x-56)$

    $0 \le x < 360 \implies 0 \le 3x < 1080 \implies -56 \le 3x-56 < 1024$

    let $u=3x-56$. $\cos{u} = -0.785 \implies u$ is an angle residing in quadrants II or III

    Quad II angles ...$u = \arccos(-0.785) \approx 141.72$ and the additional coterminal angles $501.72$ and $861.72$

    $3x-56= \{141.72,501.72,861.72 \} \implies x \in \{65.9, 185.9, 305.9 \}$

    Quad III angles ... $u = 360 - \arccos(-0.785) \approx 218.28$ and the additional coterminal angles $578.28$ and $938.28$

    $3x-56= \{218.28,578.28,938.28 \} \implies x \in \{91.43, 211,43, 331.43 \}$
    Attached Thumbnails Attached Thumbnails Finding solutions of an equation-994280c2-0bf5-4951-bb20-2a940b8c3ca5.png  
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