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Thread: spot the illegal operation?

  1. #1
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    spot the illegal operation?

    Hi folks,

    most problems can be done in more than one way, but all should give the same answer.

    In the following, I get 2 coordinates using method 1 and 3 coordinates when using method 2.

    for the curve $y = cos^3 x. sin x$ where $0 \le x \le \pi$ find the coordinates of the points where $\dfrac{dy}{dx} = 0$

    so, method 1

    $\dfrac{dy}{dx} = - 3 cos^2 x. sin^2 x + cos^3 x. cos x = 0$

    $ y' = cos^4 x - 3 cos^2 x. sin^2 x = 0$

    $ y' = cos^2 x ( cos^2 x - 3 sin^2 x) = 0$

    so $cos x = 0$

    or

    $cos^2 x = 3 sin^2 x \Rightarrow tan^2 x = \dfrac{1}{3} \Rightarrow tan x = \dfrac{1}{\sqrt{3}}$

    gives $x = cos^{-1}(0) \Rightarrow x = \dfrac{\pi}{2} and \dfrac{3\pi}{2}$ but only $\dfrac{\pi}{2}$ is in quad 1 and 2, so $x = \dfrac{\pi}{2}$ is the only solution.

    OR $x = tan^{-1} (\dfrac{1}{\sqrt{3}}) \Rightarrow x = \dfrac{\pi}{6}$ and $\dfrac{7\pi}{6}$ but range is only in quad 1 and 2 so solution in quad 3 is out of range.

    There is therefore only one solution which is $\dfrac{\pi}{6}$

    therefore the coordinates where $y' = 0$ are $(\dfrac{\pi}{2}, 0)$ and $(\dfrac{\pi}{6}, \dfrac{3\sqrt{3}}{16})$

    but, method 2

    $cos^2 x ( cos^2 x - 3 sin^2 x) = 0$

    I use $cos^2 x \equiv \frac{1}{2}(1 + cos 2x)$ and $sin^2 x \equiv 1 - cos^2 x$ and get:

    $cos 2x = - 1$ or $cos 2x = \frac{1}{2}$

    and this gives a third coordinate at $(\dfrac{5\pi}{6}, - \dfrac{3\sqrt{3}}{16})$ which is the right answer.

    so what is wrong with method 1?
    Last edited by s_ingram; Sep 2nd 2017 at 11:55 PM.
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  2. #2
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    Re: spot the illegal operation?

    Quote Originally Posted by s_ingram View Post
    ... so, method 1

    $\dfrac{dy}{dx} = - 3 cos^2 x. sin^2 x + cos^3 x. cos x = 0$

    $ y' = cos^4 x - 3 cos^2 x. sin^2 x = 0$

    $ y' = cos^2 x ( cos^2 x - 3 sin^2 x) = 0$

    so $cos x = 0$

    or

    $cos^2 x = 3 sin^2 x \Rightarrow tan^2 x = \dfrac{1}{3} \Rightarrow \color{red}{tan x = \dfrac{1}{\sqrt{3}}}$
    correction ...

    $\tan{x} = \pm \dfrac{1}{\sqrt{3}} \implies x=\dfrac{\pi}{6} \text{ or } x=\dfrac{5\pi}{6}$
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  3. #3
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    Re: spot the illegal operation?

    Ah! I see it now.

    Thanks again skeeter!
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  4. #4
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    Re: spot the illegal operation?

    fyi, I would have set the second factor equal to zero and solved as follows ...

    $\cos^2{x} - 3\sin^2{x}=0$

    $(1-\sin^2{x})-3\sin^2{x} = 0$

    $1-4\sin^2{x}=0$

    $(1-2\sin{x})(1+2\sin{x})=0 \implies \sin{x}=\pm \dfrac{1}{2}$

    ... same two solutions from quads I and II
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  5. #5
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    Re: spot the illegal operation?

    yep, tricks of the trade skeeter. Tricks of the trade!

    many thanks.
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