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Math Help - Equation for Intersection Points of Graphs

  1. #1
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    Equation for Intersection Points of Graphs

    The graphs of f(x) = 2sin(x) - 1 (blue) , and g(x) = 3cos(x) + 2 (red)
    are shown below:

    https://www.virtualhighschool.com/co.../images/38.jpg

    What equation would have the intersection points of the graphs as its solutions?

    I know that I need to set the equations equal to one another:

    2sin(x) -1 = 3cos(x) + 2
    2sin(x) -1 -2 = 3cos(x)
    2sin(x) -3 = 3cos(x)

    I think I need to get all of the trig functions in terms of the same ratio, but I'm not sure how to do that. Nor am I sure where to go from there.

    Any help would be greatly appreciated. This is my last trig question and I'm finally done the unit!

    Thanks
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  2. #2
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    Okay, I think I have gotten a little bit further, but I'm still stuck....

    I have done the following:

    2sin(x) -3 = 3cos(x)
    2sin(x) -3 = 3 /sqrt [1-sin^2(x)]
    [2sin(x) -3][2sin(x) -3] = 9 [1-sin^2(x)]
    4sin^2(x) -6sin(x) - 6sin(x) + 9 = 9 - 9sin^2(x)
    13sin^2(x) - 12sin(x) = 0
    sin(x) [13sin(x) - 12] = 0

    sin(x) = 0
    Quadrants- I, II, III, and IV
    Reference Angles= 0, 180, 360
    x= 0, 180, 360

    13sin(x) - 12 = 0
    13sin(x) = 12
    sin(x) = 12/13
    x = 67.38
    Quadrants- I and II
    Reference angle= 67.38

    0+67.38 = 67.38
    180-67.38 = 112.62

    x= 67.38 and 112.62

    Therefore, x= 1, 67.38, 112.62, 180, and 360

    But those values don't answer the question. The question asks to find what equation would have the intersection points of the graphs as its solutions?

    So, I'm still lost....
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  3. #3
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    Hello, starshine84!

    \text{Consider the graphs of: }\,f(x) \:=\: 2\sin x - 1\,\text{ and }\,g(x) \:=\: 3\cos x + 2

    \text{What equation would have the intersection points of the graphs as its solutions?}


    \text{I know that I need to set the equations equal to one another:} . Yes!

    . . 2\sin x -1 \:=\: 3\cos x + 2 \quad\Rightarrow\quad 2\sin x  -3 \:=\: 3\cos x

    Square both sides: . (2\sin x - 3)^2 \;=\;(3\cos x)^2

    . . . . . . . . . . 4\sin^2\!x - 12\sin x + 9 \;=\;9\cos^2\!x

    . . . . . . . . . . 4\sin^2\!x - 12\sin x + 9 \;=\;9(1-\sin^2\!x)

    . . . . . . . . . . 4\sin^2\!x - 12\sin x + 9 \;=\;9 - 9\sin^2\!x

    . . . . . . . . . . . . 13\sin^2\!x - 12\sin x \;=\;0

    . . . . . . . . . . . \sin x(13\sin x - 12) \;=\;0


    And we have:

    . . \sin x \:=\:0 \quad\Rightarrow\quad x \:=\: 180^on

    . . 13\sin x - 12 \:=\:0 \quad\Rightarrow\quad \sin x \:=\:\tfrac{12}{13} \quad\Rightarrow\quad x \:=\:\sin^{-1}\left(\tfrac{12}{13}\right)
    . . . . x \:=\:\begin{Bmatrix}67.38^o + 360^on \\ 112.62^o + 360^on \end{Bmatrix}


    Since squaring an equation often introduces extraneous roots,
    . . we must check our results.


    We find that the solutions are:

    . . \begin{Bmatrix}x \;=\;180^o + 360^on \\ \\[-4mm] x \;=\;112.62^o + 360^on\end{Bmatrix}\;\text{ for some integer }n

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  4. #4
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    I think I understand... thanks!
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