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Math Help - Angle between curves in parametric form

  1. #1
    DBS
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    Exclamation Angle between curves in parametric form

    Hi,

    In a problem i have to find the angle between two curves that are in parametric form. Those curves are:

    (e^t*cos(t),e^t*sin(t)) and (R*cos(s),R*sin(s)) where s,t are in [0,2*Pi] and R>0

    I can't even find their intersection point. I tried something but i had to exclude two values of s and t which are Pi/2 and 3*Pi/2 so i could obtain s=t

    Thank you in advance!
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  2. #2
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    Hello, DBS!

    I've made a little progress . . .



    I have to find the angle between two parametric curves:

    . . \begin{Bmatrix}x &=& e^t\cos t \\ y &=& e^t\sin t \end{Bmatrix} \qquad \begin{Bmatrix}x &=& R\cos s \\ y &=& R\sin s\end{Bmatrix}\quad \text{ where }s,t \in [0,\,2\pi]\:\text{ and }\:R>0

    I can't even find their intersection point.

    Equating x's and y's:

    . . \begin{array}{ccccccccc}<br />
e^t\cos t &=& R\cos s & \Rightarrow & e^t &=& R\,\dfrac{\cos s}{\cos t} & [1] \\ \\[-3mm]<br />
e^t\sin t &=& R\sin s & \Rightarrow & e^t &=& R\,\dfrac{\sin s}{\sin t} & [2]\end{array}


    Equate [1] and [2]: . R\,\frac{\cos s}{\cos t} \:=\:R\,\frac{\sin s}{\sin t} \quad\Rightarrow\quad \sin s\cos t - \cos s\sin t \:=\:0

    We have: . \sin(s-t) \:=\:0 \quad\Rightarrow\quad s-t \:=\:\begin{Bmatrix}0 \\ \pi \end{Bmatrix}

    . . Hence: . s \;=\;\begin{Bmatrix}t \\ t+\pi \end{Bmatrix}

    It turns out that s \:=\:t +\pi is an extraneous root.

    Hence, the intersection occurs at s \,=\,t


    Then [1] becomes: . e^s \:=\:R\,\frac{\cos s}{\cos s} \quad\Rightarrow\quad e^s \:=\:R \quad\Rightarrow\quad s \:=\:\ln R


    Therefore, the curves intersect at: . \bigg(R\cos(\ln R),\;R\sin(\ln R)\bigg)


    But check my work . . . please!
    .
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  3. #3
    DBS
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    Thanks very much!

    Actually what i find weird is that i got to s=t with a method but the problem is for that i got to tan(t)=tan(s) but i had to exclude the values that i told earlier...but your method seems correct enough without excluding values. So again, thank you.
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  4. #4
    DBS
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    Where can i learn to write like you did?...is it with latex by any chance? thx
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