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Math Help - Solving a trig equation.

  1. #1
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    Solving a trig equation.

    solve for x


    cot-1 (4x) + tan-1 (x) = 60 degrees

    its so hard
    Last edited by mr fantastic; August 24th 2010 at 02:22 PM. Reason: Re-titled.
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  2. #2
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    Quote Originally Posted by aeroflix View Post
    solve for x


    cot-1 (4x) + tan-1 (x) = 60 degrees

    its so hard
    Let cot^(-1) (4x)= w

    4x = cot w

    tan w = 1/4x

    Let also tan^(-1) x =p

    x= tan p

    so w + p=60

    tan (w+p)= tan 60

    (tan w+tan p)/(1-tan p tan w)= tan 60

    Can you take it from here?
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  3. #3
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    its kinda confusing. whats your point
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  4. #4
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    Hello, aeroflix!

    The answers are truly ugly . . . but they check out!

    I'll baby-talk through this for you . . .


    \text{Solve for }x\!:\;\;\cot^{-1}(4x) + \tan^{-1}(x) \:=\: 60^o .[1]

    Let \theta \,=\,\cot^{-1}(4x)

    . . Then: . \cot\theta \,=\,4x \,=\,\frac{adj}{opp}

    So  \theta is in a right triangle with: . opp = 1,\;adj = 4x

    . . \tan\theta \,=\,\frac{1}{4x} \quad\Rightarrow\quad \theta \:=\:\tan^{-1}\left(\frac{1}{4x}\right)

    Hence: . \cot^{-1}(4x) \:=\:\tan^{-1}\left(\frac{1}{4x}\right)


    Then [1] becomes: . \tan^{-1}\left(\frac{1}{4x}\right) + \tan^{-1}(x) \;=\;60^o .[2]



    We need this identity: . \tan(A + B) \;=\;\dfrac{\tan A + \tan B}{1 - \tan A\tan B}


    Take the tangent of both sides of [2]:

    . . \tan\bigg[\tan^{-1}(\frac{1}{4x}) + \tan^{-1}(x)\bigg] \;=\; \tan(60^o)

    . . \dfrac{\overbrace{\tan\left[\tan^{-1}(\tfrac{1}{4x})\right]}^{\text{This is }\frac{1}{4x}} + \overbrace{\tan\left[\tan^{-1}(x)\right]}^{\text{This is }x}}{1 - \underbrace{\tan\left[\tan^{-1}(\tfrac{1}{4x})\right]}_{\text{This is }\frac{1}{4x}} \underbrace{\tan\left[\tan^{-1}(x)\right]}_{\text{This is }x}} \;=\;\sqrt{3}


    So we have: . \dfrac{\frac{1}{4x} + x}{1 - \frac{1}{4x}\cdot x} \;=\;\sqrt{3}

    . . which simplifies to: . 4x^2 - 3\sqrt{3}\,x + 1 \;=\;0


    \text{Quadratic Formula: }\;x \;=\;\dfrac{3\sqrt{3} \pm\sqrt{11}}{8} \;\approx\;\begin{Bmatrix}1.064097152 \\ 0.234940954 \end{array}

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