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Finding a value of theta for which a tangent line is horizontal

  1. May 7, 2008 #1
    1. The problem statement, all variables and given/known data

    Find the smallest positive value for `theta` for which the tangent line to the curve `r = 6 e^(0.4 theta)` is horizontal

    3. The attempt at a solution

    I tried to use the (r'sin(theta)+rcos(theta))/(r'cos(theta) - rsin(theta)) formula to get the tangent line, which I got to be...


    divided by


    And then find the theta that would make this equation 0, got pi/2, but the answer is 1.95130270391. Help?
  2. jcsd
  3. May 8, 2008 #2


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    Just a quick observation, but all of the 6e^(.4x)'s will cancel out leaving .4*sin(x)+cos(x) on top and .4cos(x) - sin(x) at the bottom.
  4. May 8, 2008 #3
    Hokay, what I did to solve this was find y as a function of [tex]\vartheta[/tex], then differentiate with respect to [tex]\vartheta[/tex], and set equal to 0, then solve for [tex]\vartheta[/tex].

    so you have r = 6e^(0.4[tex]\vartheta[/tex])

    and y = rsin([tex]\vartheta[/tex])

    so y = 6sin([tex]\vartheta[/tex])e^(0.4[tex]\vartheta[/tex])


    dy/d[tex]\vartheta[/tex] = 6cos([tex]\vartheta[/tex])e^(0.4[tex]\vartheta[/tex]) + 2.4sin([tex]\vartheta[/tex])e^(0.4[tex]\vartheta[/tex])

    = 0

    factor out the exponential term.

    The exponential term cannot be equal to 0 so divide it out.

    Now you have

    6cos([tex]\vartheta[/tex]) + 2.4sin([tex]\vartheta[/tex]) = 0

    since sin([tex]\vartheta[/tex]) = 1 - cos^2([tex]\vartheta[/tex])

    you get the quadratic

    -2.4cos^2([tex]\vartheta[/tex]) + 6cos([tex]\vartheta[/tex]) + 2.4 = 0

    solving you get

    cos([tex]\vartheta[/tex]) = -0.3508 or 2.851

    2.851 is not in the range of cos (wtf?)

    -0.3508 is, so take the arccosine of that.

    you get [tex]\vartheta[/tex] = 1.93 + N2[tex]\pi[/tex] where N is an integer

    Although my answer is off by 2 hundredths of what you say it is.... how did you come up with 1.95?
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