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Math Help - arctan and arccot

  1. #1
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    arctan and arccot

    why does -3 cot^{-1} \frac{3}{\sqrt{x^2-9}} = 3 tan^{-1} \frac{3}{\sqrt{x^2-9}} ??
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  2. #2
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    Re: arctan and arccot

    I get their difference is

    \begin{align*}&\frac{3\pi}{2} \;\; \{x: \left|x\right| < 3 \} \\ \\ &\frac{-3\pi}{2} \;\; \{x: \left|x \right| \geq 3 \} \end{align}

    so I wouldn't call them equal. Why do you think they are?
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  3. #3
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    Re: arctan and arccot

    because \int \frac{\sqrt{x^2-9}}{x} dx = \sqrt{x^2 - 9} - 3 cot^{-1} \frac{3}{\sqrt{x^2-9}} and \sqrt{x^2 - 9} +  3tan^{-1}\frac{3}{\sqrt{x^2-9}}
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    Re: arctan and arccot

    Quote Originally Posted by Jonroberts74 View Post
    because \int \frac{\sqrt{x^2-9}}{x} dx = \sqrt{x^2 - 9} - 3 cot^{-1} \frac{3}{\sqrt{x^2-9}} and \sqrt{x^2 - 9} +  3tan^{-1}\frac{3}{\sqrt{x^2-9}}
    Ok, I show that the derivatives of each of these expressions do indeed equal the integrand but remember that there is a constant of integration when you integrate. So the two expressions you have that integral equal to can in fact differ by a constant, which they apparently do.
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  5. #5
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    Re: arctan and arccot

    yeah I left out the constant because it's exactly that, some constant that they vary by.
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  6. #6
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    Re: arctan and arccot

    Given any right angle triangle with legs A and B and hypotenuse C it's pretty easy to show that: \tan^{-1}(B/A) = \frac {\pi} 2 - cot^{-1}(A/B). Hence  3 \tan^{-1}(B/A)+ 3 cot^{-1}(A/B) = \frac {3 \pi} 2.
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  7. #7
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    Re: arctan and arccot

    Aren't arctan (B/A) and arccot(A/B) the same angle ?
    Last edited by BobP; February 13th 2014 at 12:22 PM.
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  8. #8
    MHF Contributor ebaines's Avatar
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    Re: arctan and arccot

    Quote Originally Posted by BobP View Post
    Aren't arctan (B/A) and arccot(A/B) the same angle ?
    My bad, thanks for catching the error. What I should have wriiten is:

    it's pretty easy to show that: \tan^{-1}(B/A) = \frac {\pi} 2 - cot^{-1}(B/A). Hence  3 \tan^{-1}(B/A)+ 3 cot^{-1}(B/A) = \frac {3 \pi} 2
    Thanks from romsek
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